A word Americans use to describe an English or British person. Comes from when British sailors used to eat limes to avoid scurvy. Although some entries have described the phrase as derogatory, it's really not that offensive at all, certainly nothing like the N-word, for example. It depends on the context, if the American uses it affectionately, the Briton should have no reason to be offended. If the American uses it while hurling abuse and expletives at the Briton, than their whole tone is what's offensive rather than the use of the word "limey".
American: Hey there limey, how's it going in England?
Englishman: Fine thanks, are you alright?
Cab Driver: What, no tip? You cheap limey cocksucker!
Englishman: Well in Britain we don't tip EVERYONE, you Yank twat!