That being said, there can be a few things that go into apologizing. We try to allow grace in any situation. Adults and children mess up just the same. I also try to keep in mind that both parties may need a cooling off period before an apology takes place, so don't force an apology.
Another thing I've kept in mind is that I can't step in to every single little squabble that goes on. We don't sit back and let the kids beat on each other, but there are certain times where we want them to use their problem solving skills to solve conflicts themselves. Believe me, my oldest daughter thinks her brothers can just breathe wrong sometimes. ;-)
So, here's what we try to do in our household:
1. Apologize and name the offense.
2. Offer a solution.
3. Ask for forgiveness.
Step #1 is important because it teaches the child to apologize and admit that they made the wrong choice. Sometimes this can be the hardest part. (Parents: I would also encourage you to give your child grace here. Don't berate your child over the choice they made. After they've realized it's wrong, move onto the next step. There's no sense in beating a dead horse.)
Step #2 helps the child think of a better choice for next time. This has helped our kids learn to respect others.
Step #3 gives the other child a chance to forgive and usually helps dissipate the situation quickly. I do not force my children to hug, since sometimes they are still mad. A quick hand shake can also do.
A scenario might go like this:
I'm sorry that I push past you. Next time I will try to be more patient. Will you forgive me?
I'm sorry that I accidentally hit you with the ball. I really didn't mean to. Next time I'll try to be more careful with where I throw it. Will you forgive me?
Adapting this has really worked in our household. It's taught my husband and I to give our children grace and allows us to restore peace quickly. We've also noticed that we don't need to step in as much to break up squabbles, because they've learned to apologize on their own very quickly. As our kids have gotten older, this is a really important skill to learn.
I hope this might help you on your parenting journey! What are some methods of apologizing that work well in your household?