For most things in life, I have always expected a sincere "I'm sorry" to be enough to remedy the situation. Yes of course I might still have punishment to face, the person might be mad, and it might take while before things are at ease again, but I generally expect my remorse to repair most of the damage. If I take the time to apologize and the other party is unwilling to listen, I have no problem writing them off as unreasonable or even a dick. They are certainly someone I didn't care to associate with. I've probably written off far too many people in my life because of this tendency.
But, every once in a while I am faced with a situation where "I'm sorry" doesn't fix the situation, and I know the person so well, that the way they respond to my apology is not enough to outweigh the impression I have created of them over the course of our relationship. In these instances I am forced not to look at the other person, but myself. Self doubt. Reflection. Introspection.... About my character, how I deal with other people, and how I view myself.
That's what I have been going through lately.