Fur was the name for the body hair of non-Human mammals.
One example of a creature with fur was the Tarcassian razor beast, which was a winged, furry animal noted for the spines on its wings and its loping way of locomotion. When she was a child, Guinan had an imaginary one as a pet. In her mind it had dark brown fur, gold eyes, and enormous wings that let it fly so fast no one but her could see it. (TNG: "Imaginary Friend")
When discussing tribbles in 2268, Spock noted to Dr.McCoy that while they were "nice, soft, and furry and they make a pleasant sound", a violin covered in ermine would be the same, and he would "see no advantage in having one." (TOS: "The Trouble with Tribbles")
One of the youths living at the settlement on Gaia, Gabriel, told Worf that he wanted to join the "Klingons" so that he could ride a wild torga, hunt, and trade furs for the things he needed, as the other "Klingons" did. (DS9: "Children of Time")