Many times, these are interchangeable (except, you would never say "apologize me," instead, say "I apologize") I apologize=I'm sorry, so you can use it if you bump into someone, or if you have made a mistake, or if you have to interrupt someone, etc. Forgive me is actually not very common. The only way I can really imagine people using it is within a sentence. "Please forgive me, but..." or "I cannot make it out tonight, I hope you can forgive me." or something, but it is never really used on it's own (In Texas, at least). Pardon me you can use almost all the time. Very flexible phrase on its own, although a bit formal. Saying "excuse me" is usually for when you interrupt something, are trying to politely get the attention of somebody, or are trying to move around someone.