Weber was born in Briarwood, Queens; his mother was a nightclub singer and his father was a manager of Borscht Belt comedians. Weber is a graduate of the High School of Performing Arts and the State University of New York at Purchase.
Weber started appearing in TV commercials while still in the third grade. After leaving college he became a member of the Mirror Reparatory Company and appeared opposite legendary actress Geraldine Page in several productions before winning a role as Julianne Moore's ill-tempered and ill-fated boyfriend on the CBS daytime drama, As The World Turns in 1985 - 1986. He appeared in several motion pictures or TV mini-series such as The Flamingo Kid, the 1987 film Hamburger Hill and in the acclaimed "The Kennedy's of Massachusetts" as the young JFK. Comfortable with both comedy and drama, he plays "nice guys" and evil characters with equal flair.
His most well known role was as Brian Hackett, the skirt-chasing airplane pilot-brother of Joe Hackett (played by Tim Daly) on the comedy TV series, Wings, in sharp contrast to his critically praised performance in the television mini-series version of Stephen King's The Shining playing the alcoholic, murderous Jack Torrence. Several years later, Weber starred in his own short-lived half-hour comedy "Cursed", joined the cast of ABC's "Once and Again" as the tortured artist Sam Blue, and starred the next year in the acclaimed show "The D.A." once again for ABC. Weber first appeared on Broadway in Tom Stoppard's The Real Thing and in 2001-2002 took over for Matthew Broderick as Leo Bloom in the Broadway production of The Producers (musical). In 2005 he appeared alongside Kevin Spacey in London at the Old Vic's production of "National Anthems". Weber also wrote and produced 2003's "Clubland", a Showtime movie in which he and Alan Alda played father and son talent agents in 1950s New York City (for which Alda was Emmy nominated). He recently appeared in another Stephen King adaptation of You Know They Got a Hell of a Band, from the Nightmares & Dreamscapes mini-series.
He is currently set to co-star with D.L. Hughley and Matthew Perry in Studio 60 on the Sunset Strip, a new television series by West Wing creator Aaron Sorkin. He will play the role of network boss Jack Rudolph.
Weber met his first wife (whom he has since divorced), Finn Carter on the set of As The World Turns while she was playing Sierra Esteban Montgomery. On July 9, 1995 he married his second wife, interior decorator and former Los Angeles Bureau chief for MTV, Juliette Hohnen. Weber and Hohnen are the parents of two sons: Jack Alexander Hohnen-Weber (born January 5, 2001) and Alfie James (born February 25, 2003).