Conrad, the youngest of three boys, is the son of James Watson Conrad, an Engineer, and Margaret Clement Conrad, a librarian. He is the great-grandson of electrical Engineer Frank Conrad, great-nephew of Martin Withington Clement, who was President of the Pennsylvania Railroad 1935-1948, and great-nephew of Maj.General Charles M. Clement, commander of the 28th Division. He is a native of Swissvale, Pennsylvania, and grew up on the border of Edgewood, both suburbs of Pittsburgh. In the early 1980s, he transferred as a sophomore from Swissvale High School to The Kiski School, an all-boys preparatory school in Saltsburg, Pennsylvania, on a scholarship. Conrad graduated from The Kiski School in 1985. He studied history at Brown University, and began acting while he was there. He left Brown in 1990 and worked as a carpenter, a barista, a housepainter, and interviewed former steelworkers for a historical society in Pittsburgh.

Conrad made his feature film debut with a small role in the 1994 film Under Heat. In 1996, he won a role as Leo Roth on the television series Relativity; however, he was already committed to a Pittsburgh production of the Tom Stoppard play Arcadia. Because the producers felt Conrad was right for the part, production on the series was later scheduled to accommodate his work on the play. Conrad went on to appear in the 1998 film Return to Paradise with Vince Vaughn and Anne Heche.

Continuing with his stage career, Conrad made his Broadway debut as Blythe Danner's much younger lover in a revival of Terence Rattigan's The Deep Blue Sea in 1998. He went on to appear in off-Broadway productions of Troilus and Cressida, Richard II and Tom Stoppard's Indian Ink. Most recently, in June 2009, he performed as Pale in the play Burn This at the New Hazlett Theater in Pittsburgh. In 1999, Conrad appeared in the TV drama The Weekend with Gena Rowlands and Brooke Shields. That same year he appeared in the Hallmark Hall of Fame television movie A Season for Miracles, opposite Carla Gugino.

He portrayed a naval lieutenant alongside Robert De Niro and Cuba Gooding, Jr. in Men of Honor (2000) and also appeared in recurring roles on the WB series Roswell (1999–2002) and Fox’s dramatic series Boston Public (2000–2005). In 2003, Conrad appeared in a small role in the Woody Allen comedy Anything Else and was cast as Detective Ed Exley in the network adaptation of "L.A. Confidential"; though the pilot was not picked up to series, Conrad soon found work on NBC’s Miss Match, in which he tries to win the heart of professional matchmaker Kate Fox (Alicia Silverstone). The series was pulled after airing 11 of 18 episodes.

In 2005, Conrad appeared in the comedy Wedding Crashers with Vince Vaughn and Owen Wilson, as well as landing a regular role on the CBS series Ghost Whisperer, in which he portrayed Jennifer Love Hewitt’s paramedic husband, Jim Clancy until the series was canceled in 2010. In 2010 he was seen on an Episode of CSI: Miami. In 2013 he began playing the recurring role of Ian Quinn in the first and fifth seasons of Agents of S.H.I.E.L.D..

He then went on to study theatre at New York’s prestigious Juilliard School as a member of the Drama Division's Group 25 (1992–1996). While at Juilliard, Conrad appeared in a stage adaptation of John Irving's novel The Cider House Rules, written by Peter Parnell and co-directed by actor Tom Hulce. In 1995 he left Juilliard prior to completing his final year, in order to accept a role in the film Snow White in the Black Forest starring Sigourney Weaver (released in 1997 on Showtime as Snow White: A Tale of Terror).