The Four Important Steps:
1) Express remorse...
These words are at the center of the whole deal. They mean something very important, and I talk to my kids about it frequently. When you say the words "I"m sorry", you mean that you are repenting, turning away from a particular behavior, acknowledging that you were wrong and that you wronged your brother.
2) ...for a specific offense
"I'm sorry that I..."
This is the part most often left out of an apology for the purpose of "saving face." In order to truly apologize, you've got to acknowledge that there was some specific thing that you did that was a trespass against your brother. Something you did wrong. So many apologies that I've heard, given and received over the years are half-a$$ed measures that leave the person apologized to sounding like the guilty party: "I'm sorry that you got your feelings hurt" Oh puhlease. That's not an apology. There's no acknowledgment of guilt.
3) Acknowledge your need for the other person's forgiveness
"I'm sorry that I hit you, will you please forgive me?" (we're going with the most commonly heard apology around here for this example)
Once you've turned away from the sin and acknowledged your guilt, you still have to admit that you need forgiveness from the other person. In a way this validates the anger and outrage of the sinned-against. I did something wrong, something specific and you are justified in your outrage, but I'm asking you to put it aside and take me back as your friend. It puts the asker in a vulnerable position and forces a reversal in position between the two combatants. No apology is complete without this humility.
4) Forgive freely
"Yes, I forgive you."
In our house the rule is, once someone has complied with all the requirements of a proper apology, the offended person is bound to forgive-- no questioning of motives or sincerity allowed. Obviously this won't occur in every situation where my children offer an apology throughout their lives. I know there will be times when their vulnerability and humility is thrown back in their faces, when their apology is greeted with skepticism about their motives and sincerity. But for now, on the training grounds of our family life, this is extremely important. Only the Lord knows the heart of a man and in place of perfect knowledge, they are to give one another the benefit of Mercy. Even when there has been a serious infraction of household rules and some sort of correction or consequence is called for, if forgiveness is sought, then it is freely given. After forgiveness, then we deal with spankings or other consequences.
This last part is so vital for our family. Judah, in particular, with his sensitive heart has on more than one occasion wailed post-spanking, "Bu-bu-but you forgot to say I forgive you!!" Discipline administered in the atmosphere of repentance and forgiveness is SO much more effective.