early 14c., polishen "make smooth or glossy" by friction or coating (of the surface of wood, stone, metal, etc.), from Old French poliss-, present participle stem of polir (12c.) "to polish, decorate, see to one's appearance," from Latin polire "to polish, make smooth; decorate, embellish;" figuratively "refine, improve," said by Watkins to be from PIE root *pel- (5) "to thrust, strike, drive," via the notion of fulling cloth, but there are other guesses.
The figurative sense of "free from coarseness, to refine" in English is recorded from mid-14c. Compare polite. Related: Polished; polishing. To polish off "finish" is by 1829 in pugilism slang, probably from the application of a coat of polish as the final step in a piece of work.
1590s, originally figurative, "absence of coarseness, elegance or style of manners," from polish (v.). From 1704 as "smoothness of surface;" 1705 as "act of polishing;" 1819 as "substance used in polishing."
by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.
Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.