Every disagreement in my family, and the resultant aftermath, is pretty much handled in the same way. Someone mediates the dispute, each person admits their own fault and expresses understanding of the situation, asks for forgiveness as necessary, gives forgiveness as needed, and re-commits to growing from the conflict. We often discuss the details of what happened and the heart issues that precipitated the conflict. The primary goal is for each person to accurately understand their own heart and what led to the conflict.
As the father, I have the hardest position, because I normally have to mediate my own errors which means that I have to start by admitting what I did wrong first, and I need to do that without any reference to any fault on the part of my children, or wife. After I have admitted my fault and have made sure that I haven't missed anything I have done wrong, then I can work with the other person to see what they did wrong. (And sometimes I leave that step out, because placing blame and finding punishment is never the goal of these situations.) Some times my wife mediates between me and the kids, but normally if it is something as simple as me losing my temper and yelling, I go on my own right away.
We never use the motif, "I am sorry." Although we certainly are sorry, the real issue is the heart issue that caused the conflict. In other words, it isn't enough for me to say to my daughter, "I am sorry I yelled at you." That merely reflects my regret for the way the situation ended up. The correct communication is, "I was wrong, I was angry at you. Will you forgive me for treating you wrong?" True reconciliation of relationship can't really happen on the basis of feeling sorry, it can only happen on the basis of an honest admission of offense to the person who has been hurt. Fortunately, in my family, we all understand that life is like a bag of rocks that a runner is carrying - we always bump up against each other and we decide, in advance, to understand, forgive, and move on...