Weber was born in Queens, New York. His mother, Fran (née Frankel), was a nightclub singer, and his father, Stuart Weber, was a nightclub performer and manager of Borscht Belt comedians. Weber embraces his Jewish heritage despite not having received a formal religious education. Weber graduated from Manhattan's High School of Performing Arts (1979) and the State University of New York at Purchase.
Weber started appearing in TV commercials in the third grade. After leaving college, he became a member of the Mirror Repertory Company and appeared opposite legendary actress Geraldine Page and Mason Adams in "Paradise Lost" 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 and TV mini-series, such as The Flamingo Kid, Hamburger Hill, and the acclaimed The Kennedys of Massachusetts (as the young John F. Kennedy).
His best-known role is as Brian Hackett, a skirt-chasing airplane pilot on the sitcom Wings. Several years later, Weber starred in his own short-lived half-hour comedy Cursed (later renamed The Weber Show), 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., also for ABC. Weber also had lead roles in the 1990s hit movies Single White Female and Jeffrey.

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