As others have said, it does depend on who it is, what they did and their attitude about it.
But, regardless, I try to forgive just for myself - even if we never reestablish the relationship. Carrying a grudge is exhausting!
When I forgive, I forget in the sense that I will not bring it up to the person ever again - not even in anger. Once forgiven, it's off the table for future anger.
Now I may not forget the lesson I learned. For example, if a close friend were to break my confidence and divulge some personal, embarrassing information - and later apologize and ask forgiveness... I'd forgive them, and forget in the sense that I wouldn't bring it up to them to make them feel bad constantly, but I would remember the lesson and maybe not talk to them about such personal matters in the future.