In the video, posted by the Washington Post, Mr Trump is heard telling TV host Billy Bush "you can do anything" to women "when you're a star".
The New York businessman bragged about trying to have sex with a married woman as well as kissing and groping others.
Mr Trump quickly dismissed it as "locker room banter" and added, "Bill Clinton has said far worse to me".
The clip was part of unaired footage of an Access Hollywood segment ahead of Mr Trump's appearance on the soap opera Days of Our Lives.
"I moved on her and I failed. I'll admit it," Trump is heard saying. "She was married. And I moved on her very heavily."
"I moved on her like a bitch, but I couldn't get there. And she was married. Then all of a sudden I see her, she's now got the big phony tits and everything. She's totally changed her look."
Later in the conversation, he told Bush he is "automatically attracted to beautiful" women and often tries to kiss them.
"I just start kissing them," he said. "I don't even wait. And when you're a star they let you do it. You can do anything."
"Grab them by the pussy," Mr Trump says. "You can do anything."
Mr Trump issued a statement shortly after the footage was leaked.
"This was locker room banter, a private conversation that took place many years ago," the statement read.
"Bill Clinton has said far worse to me on the golf course - not even close. I apologise if anyone was offended."
Hillary Clinton described the comments as "horrific".
"We cannot allow this man to become president," she posted on Twitter.
Donald Trump has often found himself in hot water for public comments he's made about women over the years. It turns out he's said lewd and disparaging things in private as well. Go figure.
The videotape release comes at a most inopportune time for the Republican nominee, who was trying to use his running mate's well-received debate performance on Tuesday to reboot his campaign after a week of distractions and controversy. On Wednesday he told a local news interviewer that his previous offensive comments about women - which have dogged his campaign since the first Republican primary debate last August - were made for the "purpose of entertainment".
That explanation doesn't fit with the boorish, newly married Trump shown on the video privately boasting about his efforts to seduce a married woman and have his way with whomever he pleases.
Now Mr Trump will enter Sunday's debate with a new cloud hanging over his candidacy. It's almost certain that one of the town hall participants will ask him about it. There may be no easy way to respond - but Mr Trump will have to find an acceptable answer. If he bungles it, everything else he says during the 90-minute debate won't matter.