Speaking on the final night of the party’s convention in Philadelphia, the first woman nominated by a major party said there were huge challenges.
“Powerful forces are threatening to pull us apart,” she said.
Mrs Clinton accused her Republican opponent in November’s election, Donald Trump, of sowing discord.
“He wants to divide us – from the rest of the world, and from each other.”
But Mr Trump tweeted that the speech had failed to address the threat posed by radical Islam, making the former secretary of state unfit to lead the country.
Before taking the stage, Mrs Clinton’s daughter Chelsea shared personal memories of her mother.
“My wonderful, thoughtful, hilarious mother,” the 36-year-old said, adding: “She was always there for me.”
How did Clinton do? Anthony Zurcher, BBC News, Philadelphia
Mrs Clinton had to present herself to the nation as the newly crowned Democratic Party nominee. She had to address doubts some Americans have expressed about her character.
And she had to follow up on a Wednesday night that was full of rhetorical all-stars – including Vice-President Joe Biden and, most notably, President Barack Obama.
Sometimes making the case for oneself is significantly harder than singing the praises of another.
In an acceptance speech that occasionally soared and sometimes trudged along, she did her best to frame the upcoming general election race in her favour.
She harkened to the days of the nation’s founding – a proven rhetorical tactic that Mr Trump neglected in his acceptance speech last week.
The US, she asserted, was founded on compromise and embraced “the enduring truth that we are stronger together” – wrapping her campaign slogan in revolutionary-era regalia.
After embracing her daughter, the former secretary of state delivered a speech which featured a stark admission about the threats to national unity.
“Bonds of trust and respect are fraying. And just as with our founders there are no guarantees. It’s truly is up to us. We have to decide whether we’re going to work together so we can all rise together.”
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She added: “We are not afraid. We will rise to the challenge, just as we always have.”
The risk to American prosperity included inequality, limited social mobility, political gridlock, “threats at home and abroad” and frustration over wage stagnation, she said.
Campaign pledges she mentioned:
- Creating more jobs with rising wages and pay equality
- Campaign finance reform and protecting voting rights
- Addressing climate change through clean energy policy
- “Building a path to citizenship” for undocumented immigrants in the US
- Raising the national minimum wage
- “Common-sense” gun control
- Rejecting “unfair trade deals” and supporting “homegrown manufacturers”
However, she was confident these challenges could be overcome with the American values of “freedom and equality, justice and opportunity”.
She acknowledged that too many Americans had been “left behind” by economic forces and addressed them directly: “Some of you are frustrated – even furious. And you know what? You’re right.”
Clinton’s attacks on Trump
- “We will not build a wall. Instead, we will build an economy where everyone who wants a good paying job can get one. “
- “Don’t believe anyone who says: ‘I alone can fix it.'”
- “He spoke for 70-odd minutes – and I do mean odd. And he offered zero solutions.”
- “It’s just not right that Donald Trump can ignore his debts, but students and families can’t refinance theirs.”
- “Explain to me what part of America First leads him to make Trump ties in China, not Colorado.”
Another highlight at the convention on Thursday was when the father of a fallen Muslim soldier challenged Donald Trump over his Muslim ban, prompting an ovation.
Mrs Clinton’s high-stakes remarks on the closing night of the four-day convention followed a rousing speech by US President Barack Obama.
He said on Wednesday there had never been a man or woman more qualified than Mrs Clinton to serve as president.
Mrs Clinton and Mr Trump are set for an election battle widely considered to be a tight race when voters head to the polls in November.