Democrats formally nominated Joseph R. Biden Jr. for the presidency on Tuesday night, anointing him as their standard-bearer against President Trump with an extraordinary virtual roll call vote that showcased the cultural diversity of their coalition and exposed a generational gulf that is increasingly defining the party.

Denied the chance to assemble in Milwaukee, Democratic activists and dignitaries cast their votes from locations across all 50 states and from the American territories and the District of Columbia; from the Edmund Pettus Bridge in Selma, Ala., to the iconic welcome sign in Las Vegas; and far beyond to the shores of Guam, “where America’s day begins.” They offered a grand mosaic of personal identities and experiences, many speaking in raw terms about their personal aspirations and adversities.

The second night of the Democratic National Convention straddled themes of national security, presidential accountability and continuity between the past and future leaders of the party. Like the opening night on Monday, it took the form of a kind of political variety show. Hosted by the actress Tracee Ellis Ross, the program skipped between recorded tributes from political luminaries, personal testimonials from activists and voters, and various forms of music and entertainment.

Two tributes by Republicans carried particular symbolic weight for a Democratic candidate seeking to appeal across party lines: Colin Powell, the retired general and former secretary of state in the George W. Bush administration, delivered a message of support for Mr. Biden, whom he had previously endorsed. And Cindy McCain, the widow of Senator John McCain, appeared in a video about Mr. Biden’s relationship with her husband.

By voting to nominate Mr. Biden, Democrats delivered to the former vice president a prize he has pursued intermittently since before the night’s most prominent young speaker, Representative Alexandria Ocasio-Cortez, was born. Two previous presidential campaigns ended in abrupt defeat: A plagiarism scandal extinguished his hopes in 1988, and his next effort in 2008 fizzled against the higher-wattage candidacies of Hillary Clinton and former President Barack Obama.

When Mr. Biden opted not to run for president in 2016, it was widely assumed that his dream of the Oval Office was finished. Instead, Mr. Biden’s long-awaited victory is a triumph of personal and political endurance, representing the apex — so far — of a slow upward climb by a man who entered the Senate in 1972 at the age of 30 as a grieving single father. No other presidential candidate in modern times has endured such a long interval between assuming a first major office and being nominated for the presidency.

The Tuesday night speaking lineup for the Democratic convention was always intended as a muscular contrast on foreign policy and diplomatic integrity, presented to viewers under the evening’s unsubtle theme: “Leadership Matters.”

There were two former commanders in chief, Bill Clinton and Jimmy Carter, and a former chairman of the Joint Chiefs turned chief diplomat: Colin Powell. There was Sally Q. Yates, the former deputy attorney general.

John Kerry, the former secretary of state who negotiated the Iran deal that Mr. Trump decimated, said that “America deserves a president who is looked up to, not laughed at.”

Earlier in the day, he sent a fund-raising email saying that Mr. Biden could “begin the hard work of putting back together the pieces of what Donald Trump has smashed apart.”

But putting back the pieces is probably not a feasible option:

The relationship with China has turned poisonous. Mr. Biden’s party, still reeling from Russia’s election interference in 2016, has become more hawkish on dealing with Moscow than Republicans who once cast themselves as the party of national security. North Korea has turned a project to build a few bombs into an arsenal that rivals India’s and Pakistan’s, and reconstituting the Iran deal, if that is even possible, is unlikely to change the fundamental tensions dividing the Middle East.

Mr. Biden has offered few detailed policy plans to address how he would tackle this very changed world. Instead the broad message of the virtual convention came down to this: Trust a man who ran the Foreign Relations Committee, who participated in the decisions to take out Osama bin Laden with a commando strike and Iran’s nuclear centrifuges with a cyber strike, and who would arrive at the White House with an experienced team.

To Mr. Trump and his supporters, that is Mr. Biden’s vulnerability. They say he stands for the establishment foreign policy that the current administration took office to destroy.

Mr. Biden, in turn, is arguing that Mr. Trump has allowed adversaries to undercut American interests, coddling strongmen, heartening the Russians and cutting deals for his friends.

Jill Biden, who as second lady of the United States taught English at a community college throughout her time in the administration, returned to a classroom to give her convention speech on Tuesday.

Dr. Biden spoke from Brandywine High School in Wilmington, Del., where she taught English in the early 1990s. She spoke from Room 232, her former classroom.

“You can hear the anxiety that echoes down empty hallways,” Dr. Biden said. “There’s no scent of new notebooks or freshly waxed floors. The rooms are dark as the bright young faces that should fill them are now confined to boxes on a computer screen.”

Dr. Biden expressed heartache over the losses from the coronavirus, as well as the frustration and fear it was inspiring among parents of schoolchildren. “Like so many of you, I’m left asking, ‘How do I keep my family safe?’” she said.

Dr. Biden’s convention speech focused in large measure on loss — something that has deeply affected her husband, Joseph R. Biden Jr., in different chapters of his life.

She recalled her early days with Mr. Biden, in the years after his first wife and daughter were killed in a car accident.

“How do you make a broken family whole?” she asked. “The same way you make a nation whole. With love and understanding, and with small acts of kindness. With bravery. With unwavering faith.”

And she recalled how Mr. Biden pressed on after the death of Beau Biden in 2015. At the time, she said, “I wondered if I would ever smile or feel joy again.” But her husband went back to work.

“Joe’s purpose has always driven him forward,” Dr. Biden said. “His strength of will is unstoppable, and his faith is unshakable. Because it’s not in politicians or political parties or even in himself — it’s in the providence of God. His faith is in you, in us.”

Dr. Biden, who married Mr. Biden in 1977, was once a reluctant political spouse. But this campaign season, she emerged as one of her husband’s most prolific surrogates, maintaining public campaign schedules at a pace that sometimes surpassed Mr. Biden’s during the in-person days on the trail early this year, and serving as a critical adviser.

When Dr. Biden finished speaking, Mr. Biden greeted her with a hug.

“Hey, everyone, I’m Jill Biden’s husband,” he said. “You can see why she’s the love of my life and the rock of our family.”

Cindy McCain, the widow of Senator John McCain, lent her voice to a video that aired as part of the Democratic National Convention on Tuesday — another nod to Republican voters who may be willing to cross party lines and support Joseph R. Biden Jr.

The video recounted the longtime friendship between Mr. Biden, a former Delaware senator, and Mr. McCain, who represented Arizona until his death in 2018.

“They would just sit and joke,” Mrs. McCain said in the video. “It was like a comedy show sometimes to watch the two of them.”

Mrs. McCain did not explicitly endorse Mr. Biden in the video. Earlier on Tuesday, she wrote on Twitter: “My husband and Vice President Biden enjoyed a 30+ year friendship dating back to before their years serving together in the Senate, so I was honored to accept the invitation from the Biden campaign to participate in a video celebrating their relationship.”

Tuesday was the second consecutive night in which the convention featured programming that could appeal to disaffected Republican voters. Monday night’s proceedings included remarks from several notable Republican figures, including former Gov. John R. Kasich of Ohio. Colin Powell, a former secretary of state in the George W. Bush administration, also appeared on Tuesday.

Mr. Biden and Mr. McCain faced each other on opposing tickets in 2008, when Mr. McCain was the Republican presidential nominee and Mr. Biden was Barack Obama’s running mate on the Democratic ticket.

Mr. McCain later would come under attack from President Trump, who as a presidential candidate in 2015 disparaged Mr. McCain’s Vietnam War service. Mr. Trump continued attacking him even after his death.

Colin Powell, a former secretary of state in the George W. Bush administration, gave a message of support for Joseph R. Biden Jr. at the Democratic convention on Tuesday night. It was his third consecutive Democratic presidential endorsement but his first appearance at one of the party’s national conventions.

Mr. Powell, who had previously said he would vote for Mr. Biden, said in his convention remarks that the country needed a “commander in chief who takes care of our troops in the same way he would his own family. For Joe Biden, that doesn’t need teaching. It comes from the experience he shares with millions of families sending his beloved son off to war.”

Mr. Powell said Mr. Biden would be a president “who unites us” domestically and would restore American alliances abroad. Drawing a contrast with President Trump, Mr. Powell said that Mr. Biden would not be swayed by “the flattery of dictators and despots.”

Shortly before Mr. Powell spoke, another former secretary of state, John Kerry, had even harsher words for Mr. Trump and his leadership at the White House, invoking a recent episode that Mr. Trump has been sensitive about, when he was briefly whisked to the basement for his safety.

“Our troops can’t get out of harm’s way by hiding in a White House bunker,” said Mr. Kerry, the Democrats’ 2004 presidential nominee. He called Mr. Trump’s trips overseas “blooper reels.”

Mr. Kerry, who helped orchestrate the 2015 Iran nuclear deal that Mr. Trump has abandoned, said that by contrast, Mr. Biden “knows that even the United States of America needs friends on this planet.”

Mr. Kerry, a decorated Vietnam War veteran, joined a group of national security officials who offered pretaped testimonials, a group that included Marie Yovanovitch, the former ambassador to Ukraine, and former Defense Secretary Chuck Hagel, who accused Mr. Trump of “dereliction of duty” for failing to ask President Vladimir V. Putin of Russia about reports that Moscow had placed bounties on American troops in Afghanistan.

Mr. Powell, a retired general who served as Ronald Reagan’s national security adviser and then as chairman of the Joint Chiefs of Staff, was once regarded as the future of the Republican Party and contemplated running against Bill Clinton in the 1996 election. Though he is a figure of the political past at this point — Mr. Powell is 83 — he may carry a certain moral weight in addressing moderate voters and others who feel alienated from a party they once called home.

Mr. Powell endorsed Barack Obama in 2008 and 2012, and Hillary Clinton in 2016.

In a convention heavily focused on racial justice and increasing minority representation in government, it is also notable that Mr. Powell, the Harlem-born son of Jamaican immigrants, broke several tall racial barriers as a Republican appointee. He was the first Black person to hold every top office he occupied: national security adviser, Joint Chiefs chairman and secretary of state.

His appearance — like that of another Republican, former Gov. John R. Kasich of Ohio, on Monday — is eased by a virtual convention that lacks a Democratic audience that might otherwise be inclined to jeer a former senior Bush administration official who helped make the case for war in Iraq in 2003.

Ady Barkan, the well-known progressive activist and supporter of single-payer health care, put a spotlight on the critical issue of health care on Tuesday night, addressing a subject that Democrats are likely to talk about up and down the ballot this fall.

“Even during this terrible crisis, Donald Trump and Republican politicians are trying to take away millions of people’s health insurance,” Mr. Barkan said. “With the existential threat of another four years of this president, we all have a profound obligation to act, not only to vote, but to make sure that our friends, family and neighbors vote as well.”

Mr. Barkan, who has A.L.S. and spoke through a computerized voice, appeared as part of a portion of the evening focused on health care, an issue that was a critical advantage for Democrats in the 2018 midterm elections and is likely to play an important role this fall as well.

Mr. Biden has campaigned on expanding health coverage under the Affordable Care Act, the signature domestic achievement of the Obama presidency. Mr. Trump and congressional Republicans sought to repeal the health law in 2017 but failed, and the Trump administration is now asking the Supreme Court to overturn the law.

Mr. Barkan took a while to come around to supporting Mr. Biden, mirroring the political journey of many other progressives. Before making an endorsement in the Democratic primary, Mr. Barkan said he wanted the nominee to be “someone other than Biden.” He eventually endorsed Senator Elizabeth Warren of Massachusetts, and then he backed Senator Bernie Sanders of Vermont. Both candidates supported “Medicare for all,” unlike Mr. Biden.

But Mr. Biden emerged as the presumptive nominee, and Mr. Barkan endorsed him in July, providing a boost to the former vice president as he seeks to improve his popularity among progressive voters who were not initially drawn to his candidacy.

In his remarks at the convention, Mr. Barkan described the existing health care system as falling far short of what Americans deserve, and called for the election of Mr. Biden.

“With a compassionate and intelligent president,” he said, “we must act together and put on his desk a bill that guarantees us all the health care we deserve.”

Joseph R. Biden Jr. of Delaware is now the Democratic Party’s 2020 presidential nominee.

Mr. Biden, 77, reached the nomination threshold of 1,991 votes around 10:20 p.m. Eastern, when the delegation from his home state delivered all of its 32 votes to him.

The camera flashed to a grinning Mr. Biden, who plucked off his mask and rose from his seat in a school library where his wife, Jill, was to deliver a speech later Tuesday night.

“Thank you from the bottom of my heart,” he said.

The roll call vote, the central event of the national convention, was drastically revamped to accommodate the constraints imposed by the coronavirus pandemic. This year, it consisted of a series of pretaped recordings of delegates listing their vote tallies, replacing the iconic and photogenic ritual of delegates shouting their state’s numbers into a hand-held microphone.

One big plus: Most of the pretaped segments were filmed outside — on a beach in California, at a firehouse in Connecticut, atop a train platform (where else?) in Delaware — bringing air, sunshine, surf and chirping crickets to a convention for which many speeches have been filmed indoors.

A handful of the delegates who called the roll for Mr. Biden were well known.

Khizr Khan, the Gold Star father and immigrant who spoke out about President Trump’s immigration policies at the 2016 Democratic convention, represented Virginia and accused the president of standing up for “white supremacists” during the 2017 unrest in Charlottesville, Va.

Mayor Muriel Bowser, who oversaw the painting of “Black Lives Matter” near the White House after federal law enforcement officials violently disrupted a peaceful protest, represented Washington, D.C.

Jamie Harrison, who is mounting a strong challenge to Senator Lindsey Graham, Republican of South Carolina and a strong Trump ally, announced the Palmetto’s State’s votes for Mr. Biden.

Senator Bob Casey, Democrat of Pennsylvania, stood in front of Mr. Biden’s childhood home in Scranton as he announced that the state had delivered 175 delegate votes to the former vice president.

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Both Mr. Biden and the runner-up, Senator Bernie Sanders, were nominated.

Representative Alexandria Ocasio-Cortez, Democrat of New York, had the official role of putting the name of the 2020 runner-up, Senator Bernie Sanders, up for the nomination on Tuesday. But Ms. Ocasio-Cortez, a first-term member of Congress, summed up the scope of the progressive movement and goals along the way.

Some progressives were frustrated that Ms. Ocasio-Cortez, one of the party’s rising stars, was granted only a one-minute speaking slot, but she packed a lot into her just over 90 seconds.

She railed against the “unsustainable brutality of an economy that rewards explosive inequalities of wealth.” She highlighted the central causes that Mr. Sanders had pushed, including “guaranteed health care, higher education, living wages and labor rights for all people.” And she frontally addressed the need to “to recognize and repair the wounds of racial injustice, colonization, misogyny and homophobia.”

She added five words in Spanish as she called to reimagine immigration and foreign policy policies to “turn away from the violence and xenophobia of our past.”

Two words that Ms. Ocasio-Cortez did not say: “Joe Biden.” She explained that to her big audience on social media soon after.

Before her appearance, Ms. Ocasio-Cortez had mocked Republican critics who had tried to make hay of her short slot.

“If I can regularly roast Trump sycophants in 280 characters or less, I can speak to progressive values in 60 secs (& maybe filibuster a few extra ),” she wrote to Bobby Jindal, the former Republican governor of Louisiana, earlier on Tuesday.

It was an accidental viral moment for Joseph R. Biden Jr., at a time when his quest for the Democratic nomination seemed to be flagging. En route to an interview last year with the editorial board of The New York Times, Mr. Biden found himself in an elevator with Jacquelyn, a 31-year-old security guard who shyly admitted she was star-struck.

“I love you,” Jacquelyn told the former vice president. “I do. You’re like my favorite.” Mr. Biden, smiling, asked if she had a camera. The two posed for a selfie on her smartphone.

It was a fleeting exchange that happened to be captured by a film crew for “The Weekly,” an FX show produced in collaboration with The Times. Mr. Biden did not receive the paper’s endorsement during the Democratic primary — that went to Senators Amy Klobuchar and Elizabeth Warren — but his easy rapport with Jacquelyn struck a chord on social media.

On Tuesday night, Jacquelyn made a more deliberate appearance before a national TV audience. The Biden campaign selected her as the first person to enter Mr. Biden’s name into nomination for president at the Democratic National Convention. She spoke before the start of the roll call vote.

“I take powerful people up on my elevator all the time,” Jacquelyn said. “When they get off, they go to their important meetings. Me? I just head back to the lobby. But in the short time I spent with Joe Biden, I could tell he really saw me, that he actually cared, that my life meant something to him.”

Her role on Tuesday was an honor that dovetailed with some of the themes the Biden team is keen to promote this week, including his support among working Americans and Black women in particular. And her appearance nods to what advisers believe is Mr. Biden’s chief political strength, a sense of empathy and ease with Americans from many walks of life.

Jacquelyn is employed by The Times as a security guard. She has no role in the newspaper’s journalism or editorial page, and she is not bound by rules that prohibit many of the company’s employees from engaging in political activity.

For Jacquelyn, who Biden aides said was declining to make her last name public, the Tuesday cameo will be the culmination of an unusual and unexpected role in a national campaign.

“I never thought I would be in a position to do this,” she told The Washington Post, which first reported on her role on Tuesday. “I never thought I was worthy enough to do this.”

Bill Clinton, once the Democratic Party’s charismatic headliner and centrist standard-bearer, played a decidedly supporting role on Tuesday night, offering a stinging rebuke of the “chaos” President Trump has brought to the office he held from 1992 to 2000.

Mr. Clinton jokingly urged voters to support Mr. Trump if “you want a president who defines the job as spending hours a day watching TV and zapping people on social media.”

The former president accused Mr. Trump of downplaying the coronavirus crisis, and of collapsing under the pressure of a real management challenge.

“At a time like this, the Oval Office should be a command center,” said Mr. Clinton, 74, speaking from his mansion in the northern suburbs of New York City. “Instead, it’s a storm center. There’s only chaos. Just one thing never changes — his determination to deny responsibility and shift the blame. The buck never stops there.”

Mr. Clinton — the last president to be impeached before Mr. Trump — described Mr. Biden as “a go-to-work president. A down-to-earth, get-the-job-done guy.”

The pandemic-imposed time constraints demanded by the crammed virtual format of the convention pulled off a feat planners of the event have been unable to accomplish in nearly three decades. Mr. Clinton spoke for only a few minutes.

His introduction of President Barack Obama at the 2012 convention in Charlotte, N.C., clocked in at 48 minutes, far longer than his allotted time slot.

Every living past Democratic president is set to speak at the Democratic National Convention this week. That makes for a sharp and intentional contrast with next week’s Republican gathering, where the party’s last living president, George W. Bush, is not expected to appear and its last presidential nominee, Mitt Romney, voted to impeach President Trump this year.

The first of three former presidents to speak was a 95-year-old Jimmy Carter, who testified that Mr. Biden has the “experience, character, and decency to bring us together and restore America’s greatness.” His wife, Rosalynn Carter, also spoke.

The Carters did not appear on camera but voiced their support for Mr. Biden over a montage of images.

Mr. Carter praised Mr. Biden as a loyal supporter of his in the Senate in the late 1970s — a sign of the longevity of Mr. Biden’s decades-long career.

“Joe Biden must be our next president,” he said.

Immediately before Mr. Carter were two links to Democratic presidential lineage: Caroline Kennedy, the daughter of President John F. Kennedy, and Jack Schlossberg, Mr. Kennedy’s grandson.

The cumulative message was an unmistakable effort to cast Mr. Biden as firmly in the Democratic mainstream, while some Republican defectors — Cindy McCain, John Kasich and Colin Powell — made the case that Mr. Trump falls well outside the G.O.P. tradition.

Senator Chuck Schumer of New York, the minority leader, offered a blunt message to viewers on Tuesday night: “America, Donald Trump has quit on you.”

Mr. Schumer addressed the convention with less than three months until November’s elections, when the Democratic Party hopes to win control of the Senate, where Republicans currently hold 53 seats.

“If we’re going to win this battle for the soul of our nation, Joe can’t do it alone,” Mr. Schumer said, standing with the Statue of Liberty behind him. “Democrats must take back the Senate. We will stay united, from Sanders and Warren to Manchin and Warner. And with our unity, we will bring bold and dramatic change to our country.”

Mr. Schumer was referring to four members of the Democratic caucus from opposite wings of the party: two well-known progressives, Senators Bernie Sanders of Vermont and Elizabeth Warren of Massachusetts, and two moderates, Senators Joe Manchin III of West Virginia and Mark Warner of Virginia.

If Joseph R. Biden Jr. wins the presidency, the fate of the Senate will have a big effect on his legislative prospects. He would face a much more challenging landscape if Republicans managed to hold on to the chamber. But Democrats have appeared in increasingly strong position, in part because of President Trump’s unpopularity.

Sally Q. Yates, the former acting attorney general who was fired by President Trump in his first month on the job for refusing to enforce a travel ban on predominantly Muslim countries, became the first of many so-called #resistance heroes on the left in the Trump era.

On Tuesday, as a speaker on the second night of the Democratic National Convention, she pressed the case that Mr. Trump, who has fashioned himself as a “law and order” president, has instead “trampled the rule of law, trying to weaponize our Justice Department to attack his enemies and protect his friends.”

She called his Muslim ban “shameful and unlawful.”

“Speaking at a political convention is something I never expected to be doing, but our democracy is at stake,” Ms. Yates said, attacking Mr. Trump for fawning over President Vladimir V. Putin of Russia, for trying to “sabotage” the Postal Service and for undermining the F.B.I. and a free press.

She said these types of moves “all have one purpose: to remove any check on his abuse of power. Put simply, he treats our country like it’s his family business. This time, bankrupting our nation’s moral authority at home and abroad.”

Ms. Yates spoke for less than five minutes, but she presented herself and other civil servants who have served in the federal government as under attack from a rogue president who has “used his position to benefit himself rather than our country.”

Ms. Yates, who was born in Georgia, has been talked about as a potential candidate in that increasingly swing state but she has so far opted against running for office.

The actress Tracee Ellis Ross, 47, who has won a Golden Globe Award for her role on “Black-ish” and moderated a book tour for the former first lady Michelle Obama in 2018, is the M.C. of the virtual convention on Tuesday.

“As a Black woman, I find myself at a crucial intersection in American politics,” Ms. Ross said. “For far too long Black female leadership in this country has been utilized without being acknowledged or valued.”

The selection of Kamala Harris as Joseph R. Biden Jr.’s running mate has started to change all that, she added.

“Hello, Kamala,” Ms. Ross said with a smile.

She was preceded by Mayor Tom Barrett of Milwaukee, who invited Democrats to come to his city after the coronavirus crisis had passed. “Unlike the president, we never made fun of face masks,” he said. “We understand why we can’t be together this week, and we hope you do too.”

The producers of “Black-ish” recently aired a 2018 episode in which the show’s characters discuss their alarm at the state of the country under President Trump. ABC, which airs the program, had been concerned the episode’s content was too political, Variety reported.

The actress and activist Eva Longoria hosted Monday night’s two-hour convention schedule; Kerry Washington will M.C. on Wednesday and Julia Louis-Dreyfus will do so on Thursday.

Mario M. Cuomo shot to Democratic Party stardom with a rousing depiction of a tale of two cities in 1984. Ann Richards brought down the house in 1988 by declaring of George H.W. Bush: “Poor George, he can’t help it. He was born with a silver foot in his mouth.” Barack Obama launched himself toward the White House in 2004 with his stirring account of “the hope of a skinny kid with a funny name who believes that America has a place for him, too.”

But in this year’s Democratic National Convention, there was not — for the first time in memory — a single keynote speaker handed the opportunity to capture the imagination of delegates and viewers at home.

Instead of designating a star for the party’s future, the Democrats assembled a mash-up of 17 of the “next generation of party leaders” to speak via video montage Tuesday night.

The list of speakers included Stacey Abrams of Georgia, who considered a 2020 presidential run herself after falling short in a bid to win her state’s governorship in 2018; three members of Congress; eight state legislators; two mayors; Jonathan Nez, the president of the Navajo Nation; and Nikki Fried, Florida’s agriculture commissioner — the only Democrat elected to statewide office in the critical battleground state.

One after another, speaking in short snippets, they offered a volley of criticism aimed at President Trump and talked up Joseph R. Biden Jr.

“You deserve more than the constant chaos that Donald Trump delivers,” said Mayor Robert Garcia of Long Beach, Calif.

Representative Conor Lamb of Pennsylvania added: “Take it from me. When you’re in the trenches, you want Joe Biden right there next to you.”

Ms. Abrams offered closing remarks, saying that Mr. Biden would be “a champion for free and fair elections.”

“Our choice is clear,” she said. “A steady, experienced public servant who can lead us out of this crisis just like he’s done before. Or a man who only knows how to deny and distract. A leader who cares about our families, or a president who only cares about himself.”

She added, “Faced with a president of cowardice, Joe Biden is a man of proven courage.”