It started some time in the 1600s I think, when Protestants started settling in Ireland.
The problems were really bad in the 70s, 80s and early 90s. IRA members detonated bombs all over the place and murdered people, including Lord Mountbatten. There were also atrocities committed by the British as well: The Bloody Sunday massacre, where soldiers opened fire on demonstrators because they thought they were under a pipebomb attack; killing 13, is the best known of these.
The BBC's news site on Northern Ireland.God, I was just looking at some Irish sources on the Irish Question. I'd love to have sympathy for them, but I really can't when all they can do is spit vitriol at the British.
Oh, and here's a message to all you Americans: Please stop funding the IRA (there is good evidence for this). It is a terrorist organisation, not some group of 'blessed defenders of freedom.'