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Bremner was born in Edinburgh, the son of two art teachers. He attended Davidson's Mains Primary School and Portobello High School. He originally wanted to be a circus clown, but was offered a chance at screen acting by television director Richard D. Brooks. One of his first notable roles was as a Glasgow schoolboy in Charles Gormley's Heavenly Pursuits (1986). He also played the lead in the BBC Scotland feature-length adaptation of the William McIlvanney short story "Dreaming" (1990).

Bremner portrayed Spud in Danny Boyle's film adaptation of Irvine Welsh's 1993 novel Trainspotting, and later Mullet, a street thug in Guy Ritchie's Snatch. In the 1994 stage version of Trainspotting, Bremner played the lead role of Mark Renton, the role played by Ewan McGregor in the 1996 film. He has played supporting roles in blockbusters such as Pearl Harbor and Black Hawk Down.

In 2017 he produced the short film No Song to Sing.

He has one daughter, with actress Marcia Rose, whom he met during the filming of Skin.