Andrews was born in Dublin. He was educated at Synge Street CBS and Presentation College, Dublin. Andrews came from a very strong republican family. His father, Todd Andrews, served in the War of Independence and the Irish Civil War for the anti-treaty side. "Todd" as he was known was also a founder-member of Fianna Fáil in 1926. Andrews's mother, Mary Coyle, was a prominent member of Cumann na mBan.
Before entering politics he worked in RTÉ as a Programme Executive. Andrews was first elected to Dáil Éireann at the 1977 general election for the Dublin County South constituency and later for the Dublin South constituency. He held his seat at each subsequent election until he decided not to contest the 1987 general election. He had been elected to the European Parliament in 1984 for the Dublin constituency and decided to concentrate on European politics instead of domestic. He remained an MEP until his retirement in 2004. He was Minister of State at the Department of the Environment in 1982.
Andrews's brother David Andrews is a former TD and Minister for Foreign Affairs. His son, Chris Andrews is a former TD for Dublin South East. One of Andrews's nephews, Barry Andrews was a Fianna Fáil TD, while another nephew, The Late Late Show's current presenter Ryan Tubridy, is a television and radio presenter with RTÉ.