Sen. Joseph Biden of Delaware, whose campaign for president has been damaged by admissions of plagiarism and embellishing his academic record, on Wednesday withdrew from the race for the 1988 Democratic nomination, saying the ''exaggerated shadow'' of his mistakes had begun ''to obscure the essense of my candidacy.''

Biden, 44, called a news conference to say he was ending his campaign with ''incredible reluctance'' and that he was ''angry at myself for having to make this choice'' between running for president and leading the fight against the nomination of Judge Robert Bork to the U.S. Supreme Court.

Biden, who is chairing the continuing Senate Judiciary Committee hearings on Bork`s nomination, said he believed seating Bork on the high court ''will be harmful for this country.''

Biden admitted ''I made some mistakes'' but that the ''exaggerated shadow of those mistakes has begun to obscure the essence of my candidacy and the essence of Joe Biden.''

''And although it`s awfully clear to me what choice I have to make, I have to tell you honestly I do it with incredible reluctance and it makes me angry. I`m angry with myself for having been put in the position-put myself in the position-of having to make this choice,'' he said.

''And I am no less frustrated at the environment of presidential politics that makes it so difficult to let the American people measure the whole Joe Biden and not just misstatements that I have made.''

But he said: ''I still believe it`s time to rekindle the spirit of idealism in this country. I believe I could, and I believe I will, make a difference'' in leaving the country better than his generation found it.

''There will be other presidential campaigns, and I`ll be there,'' he declared.

After his announcement, Biden returned to the Senate Caucus Room to lead the hearings on the Bork nomination. He was greeted with handshakes and pats on the back from senators of both parties and staff members.

Sen. Edward Kennedy (D., Mass.) opened the afternoon session by referring to Biden`s withdrawal. ''We certainly respect his decision,'' Kennedy said.

''We admire his courage in making that decision and we welcome him as chairman.''

In a chance encounter with reporters Tuesday, Biden said he still was in the race, but he acknowledged his campaign may have been ''overwhelmed'' by reports that he had used extensive quotes from other politicians without attributing them, had failed a law school course after failing to attribute a long quotation in a paper he wrote, and had inaccurately and angrily boasted about his academic record during a campaign appearance in New Hampshire.

Biden became the second Democrat this year to quit the race because of doubts raised about his character. In May, former Sen. Gary Hart of Colorado, then the front-runner, abandoned his campaign after the disclosure of his relationship with Donna Rice.

Campaign aides in Washington, Biden`s home state of Delaware and Iowa all reported that they had concluded that if Biden continued his crippled presidential campaign, he would damage both his political future and the battle to defeat Senate confirmation of President Reagan`s nomination of Bork. ''There are three considerations,'' one campaign official said. ''One is Joe himself and how does he continue, two is Bork, and three is the campaign. The consensus is that one and two are endangered by three.''

Some Democratic observers said the final blow to Biden`s chances was release of the videotape of his response to Frank Fahey, the Claremont, N.H., teacher who asked Biden on April 2 where he had gone to law school and how well he had done.

''I think I probably have a much higher IQ than you do,'' Biden angrily replied. He proceeded to misstate his academic record, but to some political strategists it was his manner that did the most damage. ''People looked at that tape and saw a man acting like a maniac,'' one Democratic political consultant said.

The Democratic presidential field has now been reduced to six-Gov. Michael Dukakis of Massachusetts, Sens. Paul Simon of Illinois and Albert Gore of Tennessee, former Gov. Bruce Babbitt of Arizona, Rep. Richard Gephardt of Missouri and Jesse Jackson-none of whom is considered a front-runner. Rep. Patricia Schroeder of Colorado is expected to announce her candidacy Monday.