Fisher was born in Glasgow, and attended Barrhead High School. He is best known for his portrayal of his character Rab C Nesbitt in the popular sitcom of the same name. He starred in the TV show Naked Video, where the Nesbitt character originated, along with the Baldy Man who also obtained a spin-off series. Fisher also worked with the late Scottish comedian Rikki Fulton on the hit sketch show Scotch and Wry. Another famous Scottish comedian he worked with was Hector Nicol in the 1979 BBC drama Just a boys'game.
He appeared in Michael Radford's 1984 film Nineteen Eighty-Four as Winston Smith's neighbour Parsons. In 2004 he played the role of Solanio in another Michael Radford film, The Merchant of Venice.
In 1994/5 he played the title role in the BBC series The Tales of Para Handy. In 2003 he appeared in the romantic comedy film Love Actually, where he played the role of the manager to fading music star, Billy Mack (Bill Nighy). Fisher had also appeared in the 1999 sitcom Brotherly Love and in the 2002 sitcom Snoddy.
He also appeared in the 2000 BBC adaptation of Gormenghast. In November 2006, he starred as Grandpa Potts in Chitty Chitty Bang Bang, in a 3 month run of the show in Edinburgh.