'apologise' originates from Late Latin and Greek via French, specifically from Greek 'apologia' meaning 'a speech in defence' combined with a verb-forming suffix.

'apologia' (Greek) passed into Late Latin/Medieval Latin as 'apologizare' and into Middle English as 'apologisen', eventually becoming the modern English 'apologise'/'apologize'.

Initially it meant 'to make a speech in one's defence' (defend a position); over time it evolved to commonly mean 'to express regret' or 'say sorry'.