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.