It is a gift to the receiver, and it is a gift/duty for the giver.
But more, it is not necessarily to come only with apology or repentance. When no apology is forthcoming, forgiveness is truly a gift, but after an apology, how much of forgiveness is duty and how much is gift?
Forgiveness is entirely a gift to the Forgiver. Forgiveness does not imply forgetting nor putting aside the act of the sinner. It is God that forgives and we must abide his will. If we don't put it aside, we carry it with us, the negativity that can stop our own spiritual growth. I wish I could think of a better phrase, but we wash our hands of the sin, as it is the sinner's duty to pay the consequences and seek atonement. Forgiveness=letting go with God.