Old English hof "hoof," from Proto-Germanic *hōfaz (source also of Old Saxon, Old Frisian hof, Old Norse hofr, Danish hov, Dutch hoef, German Huf "hof"), perhaps from PIE *kop- "to beat, strike" (source also of Sanskrit saphah "hoof," Polish kopyto "hoof;" see hatchet (n.)). But Boutkan acknowledges only Indo-Iranian cognates and writes, "We may be dealing with a typical relic form that only survived in the periphery of the IE area ...." For spelling, see hood (n.1).
A hoof differs from a nail or claw only in being blunt and large enough to inclose the end of the limb; and almost every gradation is to be found between such structures as the human nails, or the claws of a cat, and the hoofs of a horse or an ox. The substance is the same in any case, and the same as horn, being modified and greatly thickened cuticle or epidermis. [Century Dictionary]
Hoof-and-mouth disease is attested from 1887. Phrase on the hoof is from 1750 as "walking;" later it was cattlemen and butchers' slang for "not (yet) slaughtered."