"Anyone who knows me knows these words don't reflect who I am. I said it, I was wrong and I apologize," Trump said in the video released Friday night.
He said he has been changed by the travels he has made and people he has met during the campaign.
"I pledge to be a better man tomorrow and will never ever let you down," he said.
Trump also attacked Bill Clinton for what he described as abuse of women and Hillary Clinton for her treatment of her husband's alleged victims.
"Hillary Clinton and her kind have run our country into the ground. I've said some foolish things, but there's a big difference between the words and actions of other people. Bill Clinton has actually abused women and Hillary has bullied, attacked, shamed and intimidated his victims."
Trump's apology came after a day in which he faced hard criticism, including from many leaders of his own party, for a video that surfaced of comments he made on an open mic to television host Billy Bush in 2005.
In the recording, Trump tells Bush about how he tried to have an affair with an unidentified woman. Access Hollywood has identified the woman as host Nancy O'Dell.
"You know I'm automatically attracted to beautiful -- I just start kissing them. It's like a magnet. Just kiss. I don't even wait," Trump can be heard saying on the tape.
"And when you're a star they let you do it ... You can do anything."
"Whatever you want," another voice is heard saying.
"Grab them by the p----," Trump says. "You can do anything."
The recording reportedly occurred several months after Trump married his third wife, Melania.
Hillary Clinton immediately criticized the remarks, calling them "horrific."
"We cannot allow this man to become president," she said.
A slew of Republicans have also condemned the remarks, including the chairman of the Republican Party and past presidential nominees John McCain and Mitt Romney. Also criticizing him were four of Trump's GOP primary opponents, John Kasich, Jeb Bush, Marco Rubio and Ted Cruz.
"No woman should ever be described in these terms or talked about in this manner. Ever," Republican National Committee Chairman Reince Priebus said in a statement.