Conrad was born in the town of Springdale, Pennsylvania, near Pittsburgh. His family was Slovak, and there was a great deal of amateur music making at home, especially choral music. He studied music at Carnegie Institute of Technology (now Carnegie Mellon University) and at Juilliard.

He was resident composer for the Old Globe Theatre in San Diego, music director of the APA-Phoenix Repertory Company in New York, music director of the American Shakespeare Festival in Stratford, Connecticut, and served as dramaturge for the O'Neill Center in Connecticut.

He also has written numerous scores for documentary films and PBS television productions, choral and instrumental works and operas - Transformations, Black River (1975, revised 1981) and The Love of Don Perlimplin (1984) - commissioned by the Minnesota Opera Company, San Francisco Opera and Pepsico. Black River and The Love of Don Perlimplin, were written to libretti by the composer's then partner Richard Street.

Dirge from Cymbeline (1991) for men's chorus with offstage trumpet was written in memory of Conrad's lover Nikos, who died of AIDS. He has written other pieces occasioned by gay deaths. The first song of The Cricket Sings (1985), a GALA commission for the Seattle Men's Chorus, is a memorial for a young man who died of cancer.

He has written a church opera, The Wise Women (1994) for the American Guild of Organists, and an opera The Dangerous Liaisons (1994) for the San Francisco Opera. He has served as staff pianist with the Pittsburgh Symphony Orchestra and as assistant editor of Musical America magazine.

Conras has won various prizes and awards, including a George Gershwin Memorial Scholarship, two Ford Foundation Fellowships, a Gretchaninoff Prize, and a National Endowment for the Arts grant. He earned a B.F.A. from Carnegie Institute of Technology and received an M.S. from The Juilliard School, where he studied with William Bergsma, Vincent Persichetti and P.D.Q. Bach.

In 1972 he moved from New York to San Francisco; he joined the composition department at the San Francisco Conservatory in 1988, where he became Chair in 2000.

Conrad is best known for his operas and choral music, some of which is informed by his experience as a gay man. He is the first composer ever commissioned by a gay men's chorus. The New York Gay Men's Chorus (together with Susa's publisher G. Schirmer) commissioned Chanticleer's Carol (1982), an antiphonal work whose cries of "Awake!" are accompanied by brass, including an offstage trumpet.

He has since had numerous commissions by the Gay and Lesbian Association of Choruses (GALA) and its members, including the San Francisco, Boston, Minneapolis, and San Diego choruses.