Fur is the fuzzy hair on an animal like a rabbit or a raccoon. One downside of having a cat or dog is that you often end up finding their fur all over your house.
Most mammals have fur, dense soft coats that protect them from becoming too cold or hot. Northern dog breeds, for example, have a double coat of fur, with the dense bottom coat keeping them warm even in the snow. Sometimes fur is used to mean "an animal's pelt" or "clothing made from the fur of an animal." The phrase "to make the fur fly" means "to cause serious trouble."