An apology was a verbal expression of regret or remorse for an action which might have hurt or caused trouble to someone.
The Talosians refused to apologize for their capturing Captain Christopher Pike and threatening to destroy the USS Enterprise. One of the Talosians asked Pike if it was not sufficient that he had condemned their species to eventual death because of his unsuitability for their needs. (TOS: "The Cage")
In 2266, after Harcourt Fenton Mudd said what she thought were hurtful words to Spock, Eve McHuron apologized for Mudd's behavior, explaining that he was used to buying and selling people. (TOS: "Mudd's Women")
In 2370, Gul Dukat offered an apology to the people of Bajor for the existence of a labor camp on Cardassia IV that still held Bajoran prisoners, even though the Cardassians claimed to have freed all such prisoners after the end of the Occupation of Bajor. Dukat claimed that the camp prefect was to be chastised. (DS9: "The Homecoming")