Janie just said a lot of my feelings on the matter.

I tend to think of forgiveness in a couple of ways... in the traditional religious sense (wiping away all blame and concern?), and in a personal sense of simply 'not letting a former situation interfere in current interactions with a person'. As has been said, that is not the same as 'forgetting'.
Once, a guy I knew (who I already had some low opinions of) came into my house uninvited and physically assaulted another friend over a stupid matter. I threw him out and thereafter refused to associate with him... that is, I did not 'forgive' him... even when he asked someone else why.

Then... my ex-wife was careless and disposed of some personal items of mine before I could visit & retrieve them (she could easily have taken them to my mother's house.) In the 2-3 times I met with her or talked to her after that, I didn't bother to mention it. I still 'remember' and am sad about the lost items, but I don't really think it was a hateful or mean thing. Thus, I 'forgive' in the sense that, if I saw her, I'd not bring it up.

That may not totally express all senses of forgiveness, but it perhaps gives a bit of a perspective.