late 14c. "trimming or lining of a garment" (implied c. 1300 in surname Furhode "fur hood"), probably from Old French forrer, fourrer "cover with fur, line (clothing)," in general "to cover, fill with," from fuerre "sheath, scabbard" (via notion of "covering"), from Frankish *fodr or another Germanic source, from Proto-Germanic *fodram "sheath" (source also of Old Frisian foder "coat lining," Old High German fotar "a lining," German Futter, Gothic fodr "sword sheath"), from PIE root *pa- "to feed, protect."
First applied c. 1400 to the hairy pelt of an animal, whether still on the animal or not. The Old French noun might have had the sense "hide, fur, pelt" (and thus might serve as the immediate source of the English noun), but this is not attested. Absent this, the sense transfer from the lining to the material that goes to make it probably happened in English. As an adjective from 1590s.
I'le make the fur Flie 'bout the eares of the old Cur. [Butler, "Hudibras," 1663]