Old English glof "glove, covering for the hand having separate sheaths for the fingers," also "palm of the hand," from Proto-Germanic *galofo "covering for the hand" (source also of Old Norse glofi), probably from *ga- collective prefix + *lofi "hand" (source also of Old Norse lofi, Middle English love, Gothic lofa "flat of the hand"), from PIE *lep- (2) "be flat; palm, sole, shoulder blade" (source also of Russian lopata "shovel;" Lithuanian lopa "claw," lopeta "shovel, spade").
German Handschuh, the usual word for "glove," literally "hand-shoe" (Old High German hantscuoh; also Danish and Swedish hantsche) is represented by Old English Handscio (the name of one of Beowulf's companions, eaten by Grendel), but this is attested only as a proper name. Meaning "boxing glove" is from 1847. Figurative use of fit like a glove is by 1771.
"to cover or fit with a glove," c. 1400, from glove (n.). Related: Gloved; gloving. Old English had adjective glofed. Glover as a surname is from mid-13c.