I hereby forgive everyone who offended or angered me, or sinned against me (Prayer on Retiring).
Since we pray to God to forgive our mistakes, certainly we should be willing to forgive others who have offended us.
Forgiveness must be more than perfunctory. A man once heard his rabbi state that Yom Kippur would not achieve forgiveness from God unless one has forgiven others. This fellow then went over to someone he disliked and said, "I forgive you today, but I want you to know that as soon as Yom Kippur is over, I will despise you as much as before."
When we pray to God for forgiveness, we cite the verse, I have erased your sins like a thick cloud (Isaiah 44:22), which tells us how we should grant forgiveness to others - by removing all traces of resentment.
What good comes from harboring resentments? We cannot act on them, for the Torah explicitly forbids taking revenge. Since resentments have no practical purpose, and since they are obviously very negative feelings, they can do nothing more than wear down our emotions. When we find a smelly item in the refrigerator, we quickly get rid of it so that it does not contaminate the other foods. We should view negative feelings in the same light, for they can infect all our other emotions with negativity.
Forgiving others and thereby ridding ourselves of resentments is in itself not only a virtuous character trait, for it is considerate of others; more importantly, it works to our own advantage.