The 76-year-old, who has already run twice for the presidency, enters a crowded race for the 2020 Democratic nomination.
He is up against 19 other hopefuls.
They include Senators Elizabeth Warren, Kamala Harris and Bernie Sanders.
What else did he say?Mr Biden, who served as President Barack Obama's deputy for two terms, recalled President Donald Trump's much-criticised remark that there were "very fine people on both sides" of the deadly Charlottesville white nationalist riots of 2017.
- Which Democrat will take on Trump?
- Joe Biden, the touchy-feely politician
"I believe history will look back on four years of this president and all he embraces as an aberrant moment in time. But if we give Donald Trump eight years in the White House, he will forever and fundamentally alter the character of this nation, who we are, and I cannot stand by and watch that happen."
What reaction has there been?A spokesman for Mr Obama said selecting Mr Biden as his running mate "was one of the best decisions he ever made" and the two had "forged a special bond", but the former president notably stopped short of an endorsement.
A source familiar with Mr Obama's thinking said the former president believed "a robust primary in 2007 and 2008 not only made him a better general election candidate, but a better president, too".
Mr Biden, asked by reporters in Delaware why Mr Obama had not endorsed him, replied: "I asked President Obama not to endorse and he doesn't want to… whoever wins this nomination should win it on their own merits."
President Trump tweeted: "Welcome to the race Sleepy Joe. I only hope you have the intelligence, long in doubt, to wage a successful primary campaign."
What are Mr Biden's prospects?He is the most experienced of the Democratic candidates. A six-term senator, he ran twice unsuccessfully for president - in 1988 and 2008.
Mr Biden was tipped to run for president again in 2016, the year Mr Trump was elected, but ruled himself out after the death of his 46-year-old son, Beau Biden, from a brain tumour.
He has enjoyed relative popularity among Democrats in recent years, consistently leading every national poll of the Democratic primary tracked by the website RealClearPolitics.
Mr Biden is also betting on having the strongest appeal of the Democratic candidates across America's Midwest, where many low-income voters have abandoned the party in recent years in favour of Mr Trump.
But the former vice-president has been forced to address claims he inappropriately invaded the personal space of women.
He apologised, saying: "The boundaries of protecting personal space have been reset. I understand it and I'll be much more mindful."
Joe Biden enters the Democratic presidential contest as a front-runner, if not the front-runner, with huge name recognition and the potential to raise vast amounts of campaign donations.
Of course, so did Hillary Clinton in 2015 - and we all know how that turned out.
Like the former secretary of state, Mr Biden in his launch video seems to be defining himself as much by who he isn't - Donald Trump - as what he wants to do.
It was an oft-criticised strategy for Mrs Clinton in 2016, and also like Mrs Clinton, his lengthy time in the public eye is likely to be picked apart by opponents.
The former vice-president has a lot going for him. He also has a lot going against him. The durability of his campaign is one of the big questions hovering over the Democratic race.
- Joe Biden: Front-runner but with baggage
Who is Joe Biden?Joseph Robinette Biden Jr was born on 20 November 1942 in Scranton, Pennsylvania, one of four children in an Irish-Catholic family.
In 1972, Mr Biden was elected to the US Senate at the age of 29, and took office a few weeks later when he turned 30 - the minimum age to enter the Senate.
Just before he took office, he was devastated by tragedy: his wife Neilia and infant daughter Naomi were killed in a car crash.
Mr Biden first ran for the presidency in the 1988 election, but he withdrew after admitting that he had plagiarised a speech by Neil Kinnock, the leader of the Labour Party in the UK at the time.
After that bid he spent time rising through the Senate ranks, eventually becoming chairman of the judiciary and foreign relations committees.
In 2008 he ran for president again, but failed to gain the political traction he needed and dropped out again. Instead, he joined the Obama ticket.