Of course they fought Argentina for it. But recently I was surprised to find out that one of the earliest battles Horatio Nelson took part in was with the Spanish over the Falklands. So this has been going on for quite some time.
Hopefully a subject expert will drop by with a concise overview, but meanwhile, you may be interested in some previous threads
history & historical claims
attitude of the UK
a few comments sprinkled around here How "out of the blue" was the Argentine invasion of the Falklands?