late 14c., resolucioun, "a breaking or reducing into parts; process of breaking up, dissolution," from Old French resolution (14c.) and directly from Latin resolutionem (nominative resolutio) "process of reducing things into simpler forms," noun of action from past participle stem of resolvere "to loosen" (see resolve (v.)).
From the notion of "process of resolving or reducing a non-material thing into simpler forms" (late 14c.) as a method of problem-solving comes the sense of "a solving" (as of mathematical problems), recorded by 1540s, as is that of "power of holding firmly, character of acting with a fixed purpose" (compare resolute (adj.)). The meaning "steadfastness of purpose" is by 1580s. The meaning "effect of an optical instrument in rendering component parts of objects distinguishable" is by 1860. In Middle English it also could mean "a paraphrase" (as a breaking up and rearranging of a text or translation).
In mid-15c. it also meant "frame of mind," often implying a pious or moral determination. By 1580s as "a statement upon some matter;" hence "formal decision or expression of a meeting or assembly," c. 1600. New Year's resolution in reference to a specific intention to better oneself is from at least the 1780s, and through 19c. they generally were of a pious nature.