The days of Elul are days of Teshuba. We ask forgiveness from God, for any ritual transgressions we might have done. And we should seek our peer's forgiveness for any offenses or damage we might have caused them.
We also need to be willing to forgive.
In Chapter 2, Halakha 10, Maimonides discusses forgiveness. He says: "It is forbidden for a person to be insensitive (akhzari) and refuse to be appeased... rather, when someone approaches him seeking his forgiveness, he should forgive him wholeheartedly and with a positive spirit".
Forgiving is a very complex and difficult emotional task. But in the days when we prepare ourselves to ask forgiveness from God, we should be willing to forgive others as well.
Forgiving also implies the capacity of forgetting. I don't mean forgetting the bad experience and the lessons we have learned from it, but letting go the anger, the anguish and the personal feelings of revenge that an offense might have caused to us.
Emotional forgetfulness is the ultimate state of forgiveness. If we have decided to forgive an offense with our head but somehow emotionally we are still caught by hatred and resentment, then wholehearted forgiveness has not been achieved. When we don't get those negative feelings out of our system, we damage our entire emotional system and to make things worse, we give the perpetrator a free ride to take control of our minds.
I must clarify that I'm are referring particularly to social and personal offenses, i.e., when my friend, a family member, a colleague, a neighbor, etc. did something bad or said something wrong to me or about me. Political forgiveness or criminal cases are a more complicated matter which can not be addressed in these brief lines.
A person starts to live when he can live outside himself
Albert Einstein