Actually, I'll confess something here that I never have told anyone - I hate apologies for major issues. I was so indoctrinated as a kid that you must forgive someone immediately on an apology that to this day I feel manipulated when someone says "I'm sorry". (Minor things - accidentally breaking a glass or bumping into me or whatever are a whole other thing and I have no problem with them.) I would never tell anyone that because I would be horrified by making someone feel bad about apologizing, and since I logically understand that the person apologizing is sincere and not trying to force me into forgiving, but the more important the issue is to me, the less I want to hear an apology. What I prefer is that the person acknowledge what happened and then tell me it won't happen again. That's really what I want - a validation of my feelings and an assurance that they care enough to not do it again.
So, because of my neurosis about apologies, I also almost never know how to respond to them. Usually, if someone says "I'm sorry", I'll respond with, "Do you understand why I am upset?" If they say yes, then I'll say something like, "I need you to show me you understand and care by not doing this again." And then if they agree, then I'm at the point where I can say, "Thank you for understanding. I was really hurt and I appreciate knowing that you understand and it won't happen again."
I think that phrase kind of wraps up acknowledgement that they're sorry and acceptance without saying "I forgive you."