(1) When God forgives - He cancels or sets aside a required punishment for sin.
like with the atonement of Christ where forgiveness of sins is available to all who repent, except those guilty of murder or the unpardonable sin against the Holy Ghost.
(2) When people forgive each other, they treat one another with love and have no bad feelings toward those who have offended them.
It seems to me that true forgiveness is a cancellation of debt on the side of the one forgiving - but that sometimes the one being forgiven is not able to fully accept that.
Hope this helps:)