fur (n.) Look up fur at Dictionary.com
late 14c. "trimming or lining of a garment" (implied c.1300 in surname Furhode "fur hood"), probably from Old French fourrer "to line, sheathe," from fuerre "sheath, covering," from Frankish *fodr or another Germanic source, from Proto-Germanic *fodram "sheath" (cognates: Old Frisian foder "coat lining," Old High German fotar "a lining," German Futter, Gothic fodr "sword sheath").

Sense transferred in English from the notion of a lining to the thing used in it. First applied early 15c. to animal hair still on the animal.
I'le make the fur Flie 'bout the eares of the old Cur. [Butler, "Hudibras," 1663]
As a verb, from c.1300, from Old French fourrer. Related: Furred; furring.