Apologise is the British English spelling of the verb apologize. So if you need to say you're sorry for something you did in London, you should apologise.
Apologise comes from Greek word apologia, which means "a speech in defense of something." So use the verb apologise if you are making an excuse for, defending, or explaining some action. If you "accidentally" ate the last piece of pizza without offering it to anyone else and then burped in everyone's face at the table, you should probably apologise.