It used to really bother me how people would refuse to admit when they were wrong about something. I used to be warry about admitting when I was wrong about something, because upon making such an admission, I noticed that people tended to respond with disdain or some emotional equivalent of "See! I told you so! Look at how wrong you are!". It perplexed me how apologizing to someone seemed like an invitation for them to attack me.
Everything that you experience in this world is a reflection of who you are - the quality of the character that you are playing in this life. The things that you don't like about yourself don't just exist inside of your mind. They are projected into the world and manifest as situations and people you probably find rather undesirable.
When I got angry at people for refusing to admit they wronged or hurt me, when I thought I was right, I began to realize that some of that anger was rooted in my own inability to not accept people for how they are.
When another person "wrongs" you, it's very tempting to make up a story inside of your head about how it was their intention to wrong you. But the truth is that people are going to do what they are going to do, and act how they are going to act, even if other people aren't OK with it. You can't control the behavior of another person, even if what they are doing is objectively and provably wrong.
I realized that when I got angry at people for imposing themselves on me, hurting me, or otherwise not respecting me, I was refusing to accept them for how they are. I realized that that other person is just being themselves. That they treat everyone in a similar manner to how they treat me.
And who am I to invalidate someone else, just because I don't see things the way that they do? That is where I learned the importance of forgiveness...
"Forgiveness" is one of those words that have been butchered by concepts and organized religion. "Forgiveness" is a difficult word to understand because it has such a loose meaning.
How do you forgive someone? Are you supposed to let them push you around, or be OK with them harming you or not respecting your boundaries? Don't they deserve your disdain because they are acting in a way that is undesirable? Well, that depends - do you believe that you deserve to suffer simply because you've unintentionally wronged other people in the past?
Personally - I recognize that I don't necessarily deserve whatever wrongness someone else imposes on me, but at the same time I also recognize that their wrongness somehow aligns with a part of me that I don't like about myself very much, and that they are unintentionally but very effectively bringing awareness to something inside of myself that I dislike.
Forgiveness = Acceptance
You don't have to let people walk all over you to convince yourself that you are a "forgiving" person. If someone disrespects your boundaries or imposes on you, chances are that they are doing it out of their own ignorance. You can still assert your boundaries and protect your own interests when someone wrongs you, while at the same time simply accepting "this is just how this person is." And just as quickly you can forget about them and go back to focusing on things that will bring you joy and peace, rather than fixating on another person.
That is my understanding of the word Forgiveness. Simply accepting how things are or how someone else is at any given moment. There have been times where I have genuinely bothered another person without realizing it. But I knew my intentions were good, so I didn't allow myself to feel guilty about some misunderstanding that arose out of my own ignorance. So at the very least I owe that flexibility to other people when they do or say something that irks me.
And when you hold on to negative feelings that you have about other people - you are doing just that. You are holding on to an unpleasant emotion, and probably having repetitive thoughts trying to validate why you are upset with that other person. Why then, would you continue torturing yourself over something someone else is doing, that you have no control over?
You set yourself free from the ignorance of others (and of yourself), by acting with humility, and accepting people despite their flaws. As soon as you accept how others are, without holding them to an unreasonable standard of "perfection" in their dealings with you - the world will also be a little more gentle with you when you act imperfectly.
Life moves in waves. Constant expansion and contraction as you move forward through Time. If someone else being "wrong" challenges you and causes you to experience negative emotions, then you can be sure that those situations will continue happening and continue anchoring your awareness somewhere else, outside of the Present.
You overcome the "wrongness" of others (and yourself), by simply letting it Be. Don't hide it from your awareness, just say it. "Yes, I am bothered by this" and then accept it. That is the Essence of Forgiveness.