Jay Adams calls "apologizing" the world's unsatisfactory substitute for forgiveness.
"uh...sorry"..."that's ok." That is apology and it's not repentance, and it's not real forgiveness.
There is another question that parallels "what is forgiveness".
What does forgiveness DO?
What makes "forgiveness", "forgiveness"?
Is it a feeling? No - Scripture says nothing about "feeling" forgiving. Scripture tells us to forgive.
Eph.4:32 tells us, "...forgiving one another, as God in Christ forgave you."
The "as" gives us a model. Strong's gives the definition
...a)just as, even as
...b)in proportion as, in the degree that
2)since, seeing that, agreeably to the fact that
3)when, after that
What does Scripture tell us about God's forgiveness of us? Does He simply sit in heaven and "emote"? No, He gives us a promise.
Isa 43:25 "I, I am he who blots outyour transgressions for my own sake,
and I will not remember your sins.
Jer 31:34b For I will forgive their iniquity, and I will remember their sin no more."
If forgiveness were merely a "feeling", we could not rely on this promise.
If forgiveness were merely "forgetting", it would not be a purposeful relinquishing of the debt. I can "forget" my keys. But the Almighty Creator of the universe cannot forget anything. It is a commitment to not recall, not bring it to remembrance, not to bring it up, remind, mention or record the offense.
If I make that promise to forgive - to not bring it up again - and I do, that is breaking a promise. I went through a really rough time with my daughter a couple of years ago. That which was done is no longer between us - and the repentance/forgiveness process works.
We went through this book. I DID bring it up again and my daughter (as my sister in Christ) reminded me of the promise. I had my own repenting to do. The process works.
Forgiveness is a promise.