early 14c., polischen "make smooth," 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 to be from Proto-Indo-European *pel- "to thrust, strike, drive" (via the notion of fulling cloth). The sense of "free from coarseness, to refine" first recorded in English mid-14c. Related: Polished; polishing. Slang polish off "finish" is 1837, from notion of applying a coat of polish being the final step in a piece of work.
1590s, "absence of coarseness," from polish (v.). From 1704 as "act of polishing;" 1819 as "substance used in polishing."
[fr the notion of finishing a piece of work by giving it a final polish]