As the woman who got closest to the White House, Mrs. Clinton remains a singular figure in Democratic presidential politics, and a complicated one. Both beloved and blamed for her narrow loss to Mr. Trump in 2016, she retains a loyal and powerful constituency of supporters, many of whom have argued that Russian interference cost her the election.

Her re-emergence in presidential politics also serves as an implicit reminder to the Democratic left about the dangers of a divided party. While a wide array of factors contributed to Mrs. Clinton’s loss, one element was the refusal of some on the left to coalesce behind her candidacy against Mr. Trump.

For progressives now skeptical about Mr. Biden, Mrs. Clinton’s endorsement, among other things, is an invitation to reflect upon whether they want to risk four more years of the Trump presidency because they are uncomfortable with the former vice president. Mr. Biden has argued since last year that he is best positioned to defeat Mr. Trump — but that he would need a united party to do it. No one knows better than Mrs. Clinton that there is no room for error or a lack of unity among Democrats in a closely fought general election.

Mrs. Clinton’s relationship with Mr. Biden hasn’t always been easy: The two developed a respectful relationship over their decades in Washington, though one marked by slights and awkward rivalries. Mr. Biden was resentful of the attention Mrs. Clinton received when they ran against each other in the 2008 primary race, but he was the one eventually chosen as former President Barack Obama’s running mate.

They became more friendly through weekly meetings while they both served in the Obama administration. But tensions deepened after Mr. Biden considered running against Mrs. Clinton for the 2016 nomination. After her loss, he was fairly open with his critique that she had failed to talk to middle-class voters. For her part, Mrs. Clinton seriously weighed whether to run against Mr. Biden last year.

Now they will be partners again, this time on the campaign trail, as Mr. Biden tries to do what Mrs. Clinton could not. Given her three debates against Mr. Trump in 2016, and her daily strategy and fortifications dealing with his punches and counterpunches in the heat of a campaign, her counsel to Mr. Biden has the value of firsthand experience.

During the virtual event on Tuesday, Mr. Biden suggested he would continue to seek her guidance. “Hillary, I’m going to keep calling and asking for your advice,” he said.