I disagree with @DrBill. It is not about continuing as if it never happened. That is foolishness, because you leave yourself wide open to being hurt by the same thing again. Forgiveness is about letting go of your anger and pain and feelings of retribution so that you can go on with your life without thinking about how you were wronged all the time (or even occasionally), and figuring out ways to get revenge. It does not require forgetting. In fact, I believe forgetting is foolish, although we see “forgive and forget” all the time in the same sentence. I am open to further discussion about forgetting, because I’m actually curious, now, as to why people put those things in the same sentence.
@avalmez: I think you are confusing apologies with forgiveness. Your two latter examples (i can choose to realize i’ve wronged others without caring that they realize my regrets. sociopaths are persons who would are unable to recognize the impact of their actions on others and so able to get on) are about the person who does the harm, not the person who has been harmed. The person who does the harm can apologize, but they can’t forgive themselves. Only the person harmed can forgive, whether or not the person who harmed asks or not.
As to your comment that is about forgiveness (i can choose to ignore the wrongs done to me and get on without forgiving those who wronged me), I respectfully disagree. If you are ignoring the wrongs, but have not let them go (as in it still bothers you a lot and it keeps you from enjoying life to some degree), you have not forgiven. If you ignore and have let the harms go (as in not dwelling on them), then you have forgiven, whether or not you formally think that.