Apologizing
Making apologies: 
I do apologize for... I must apologize for... I apologize for... I'd like to apologize for... I am so sorry for... I shouldn't have... It's all my fault. I'm ashamed of... Please, forgive me for... Excuse me for ... I'm terribly sorry for... Please, forgive me for my... Please, accept my apologies for... Accepting apologies: That's all right. Never mind. Don't apologize. It doesn't matter. Don't worry about it. Don't mention it. That's OK. I quite understand. You couldn't help it. Forget about it. Don't worry about it. No harm is done. Examples:A: I hope you’ll forgive me for being late. I had a flat tire.
B: That’s quite all right/ I understand completely. It can happen to anyone.
A: Please accept my apologies for the mistakes in the report. I forgot to check it.
B: You don’t need/have to apologize. But next time, try to concentrate better.
A: I (would like to) apologize for shouting at you yesterday. I lost control of myself.
B: don’t worry about it. I just hope you’ll keep your temper from now on.
A: excuse me! I didn’t realize this was your seat/newspaper/…
B: That’s OK. I’m leaving anyway/I’ve finished reading it/…
A: I’m sorry. I didn’t mean to keep your book so long.
B: No problem/No big thing. I won’t need it until next semester.