1590s, "to speak in defense of;" see apology + -ize. Sense of "regretfully acknowledge" is attested by 1725. The Greek equivalent, apologizesthai, meant simply "to give an account." Related: Apologized; apologizing; apologizer.
He apologized for the many typoes
I apologized for being late
apologetics
apologia
apological
apologise
apologist
apologize
apologue
apology
aponeurosis
apophasis
apophatic