mid-13c., "an opening, an aperture;" early 15c. as "an introductory proposal, something offered to open the way to some conclusion," from Old French overture "opening; proposal" (Modern French ouverture), from Latin apertura "opening," from aperire "to open, uncover" (see overt).
The orchestral sense of "a movement serving as a prelude or introduction to an extended work" in English is recorded from 1660s.
early 15c., offre, "a proposal presented for acceptance or rejection," from Old French ofre "act of offering; offer, proposition" (12c.), verbal noun from offrir "to offer," from Latin offerre "to present, bestow, bring before" (see offer (v.)). The native noun formation is offering. Meaning "act of proposing a price to obtain or do something" is from 1540s.
c. 1300, proffren, "present oneself, appear; hand over;" mid-14c., "to make an offer or proposal," from Anglo-French profrier (mid-13c.), Old French poroffrir (11c.), from por- "forth" (from Latin pro; see pro-) + offrir "to offer," from Latin offerre (see offer (v.)). Related: Proffered; proffering. As a noun, "an offer made, something proposed for acceptance by another," from late 14c.
word-forming element in pathology meaning "condition of the blood," Modern Latin combining form of Greek haima (genitive haimatos) "blood," a word of no established etymology (replacing the usual IE word, represented in Greek by ear; possibly from uncertain PIE root *sei- "to drip" (compare Old High German seim "virgin honey," Welsh hufen), but according to Beekes this proposal "cannot explain the Greek vocalism."