Fur is a synonym for hair, used in reference to non-human animals, usually mammals; particularly those with extensive body hair coverage.--wikipedia
In mammals, fur is evolved to keep warm, to decorate, to defense, and a lot more functions when it goes awry from its origianal purpose--of course there is no "purpose" at all.
Obviously, in many circumstances fur is used for aggression, for more details and its zoopolitical implications, please see this post, which, in fact, is not sensitive but may be banned here.