(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:)

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Forgiveness is a virtue but the way people perceive it is quite relative. Some people think that certain actions are forgivable while others are not. Some people think that forgiveness encourages the wrongdoer to perform ill deeds repeatedly. Forgiveness is subjective and the act of forgiveness can have many meanings. Acceptance of apology may be forgiveness for some, while helping the other who hurt you to get out of the habit of ill-treatment may be a way for others.We all make mistakes. So when we learn to forgive others, we can also seek forgiveness when we commit follies.Our grudges will only affect the relationship with the person and not hurt the other person in any way. The ability to forgive gives us a sense of freedom and makes us suffer less and feel lesser misery and pain.