Accept the apology
We are all taught to say, "I am sorry" and apologize. As difficult as it is sometimes, we still say it, but are we able to accept someone's "sorry"? Does "sorry" warrant "forgiveness"? What does it take to forgive?
First, we should explore the elements of a true apology:
It should be specific
The apologizing party should have the "right attitude"
It should include how the behavior will not occur again (X)
Even so, there are times when an apology (a true apology) does not move us enough to forgive.
Sometimes, the hurt is just too deep, and we are not ready to let go of it. It is like an insurance policy to continue to hold on to it: by doing so, we are assuming control of the situation and hence making sure it does not happen again (this is actually an illusion). The brain has a survival mechanism that tells you to hold on to the negative. This is the negativity bias and has been engrained in the groves of your cortex for thousands of years. Holding a grudge, our brain thinks, makes us somehow ready for the next let down.
Other times, it is just too late for an apology. You already moved on and so you really don't care. Good for you.
Like many other noble acts, forgiving someone is both selfless and selfish. We all understand why it may be a selfless act: you leave someone off the hook and by doing so, you are helping that person in his or her journey. But why is forgiving a selfish act? Well, because it also eases your mind and lightens your load on your journey. It is the same as making sure your son has his coat when is cold outside: selfless because you are looking after him, and selfish because you are preventing a miserable night for you if he gets sick.
At times, it helps to frame accepting an apology from the point of view that holding a grudge is negative for your overall health. If letting your "aggressor" off the hook does not appeal to you, why not look at forgiveness as an act of self-love? At least initially.
The best part of forgiving someone is that you are once again free to move along. Other things to focus on, other people and activities to enjoy and other opportunities to grow and learn. This, in turn, makes it even easier for you to forgive: you don't have time to be marinating on the misery anymore and hence, you are more ready than ever to forgive. It is like a good vicious cycle.
If you have been forgiven in the past, it is easier for you to forgive. It is like having more empathy for those who have messed up since you were one of them before. Afterall, we are all gifted and cursed by human nature.