"It's America's basic bargain," Clinton said. "If you do your part, you ought to be able to get ahead, and when everybody does their part, America gets ahead too.
"That bargain inspired generations of American families, including my own," the former secretary of state and first lady said.
In the first major speech of her second campaign for president, Clinton portrayed herself as a fierce advocate for those left behind after the recession.
She cited President Barack Obama, and former Presidents Franklin D. Roosevelt and Bill Clinton, her husband, and said they embraced the idea that "real and lasting prosperity must be built by all and shared by all."
Her campaign said her "tenacious fighter" message will form the foundation of the 2016 White House race. She also signaled her intention to campaign on the prospect she would be the first woman elected to the White House.
Clinton told the thousands at the outdoor rally on Roosevelt Island at the start of her speech she was glad to be with them "in a place with absolutely no ceilings." She ended her remarks by saying she wanted join with her supporters to build a country where a father can tell his daughter she can grow up to be anything, "even president of the United States."
Eager and excited Democrats began assembling hours before they heard from the front-runner for the nomination. Also in the race are Sen. Bernie Sanders of Vermont, former Maryland Gov. Martin O'Malley and former Rhode Island Gov. Lincoln Chaffe.
Marc Markley of New York said he showed up at 2:30 a.m. and waited in the dark for the gates to open, with only a police officer for company.
"I was about to fall asleep earlier, but now it's totally worth it," he said. "I can't wait to get inside."
Those arriving were greeted by campaign manager Robby Mook, who took an all-hands-on-deck approach to the event by hawking merchandise - a role typically assigned to a low-level staffer or volunteer.
"Ladies and gentlemen, welcome to our rally today," Mook called out to the crowd filtering into Four Freedoms park, before directing them to a table selling campaign T-shirts and other campaign-branded gear. "Thanks for being here - and buy some products."
On Friday, the Clinton campaign released a video on Friday detailing her four decades in public service, starting with her work as a young lawyer at the Children's Defense Fund.
After the Saturday speech, Clinton planned to visit early-voting states, with events focused on her relationship with her mother and her father's background as a veteran and small businessman.
"You have to get up off the floor and you keep fighting," Clinton says in the video, discussing her failed 1993 attempt to overhaul the nation's health care system during her husband's administration. "Everyday Americans need a champion."
In her Saturday address, Clinton said, "I have been called many things by many people. Quitter is not one of them." That something that comes from her mother, Dorothy Rodham, she said, adding that she would confide in her mother after hard days in the Senate and at the State Department.
"I wish my mother could have been with us longer," Clinton said. "I wish she could have seen the America we are going to build together ... where we don't leave any one out or any one behind."
Clinton has spoken out strongly on immigration and other issues important to parts of the Democratic base. But she has been reticent on other policy questions that have divided the party, among them a trade deal with Pacific Rim nations. Obama backs it. Organized labor, liberals and others say it would cost U.S. jobs.
On Friday, dozens of union-backed House Democrats voted down a critical part of Obama's trade agenda, negotiating authority that would let him propose trade agreements that Congress could accept or reject, but not amend.
Clinton was joined by her husband and daughter Chelsea at the rally. It was the first time the family had been seen together in public since Clinton began her campaign in April, and the crowd chanted "Bill! Bill! Bill!" when she introduced him.
"Oh, that will make him so happy," Clinton said.
The Associated Press contributed to this report.