Get all emotional and say sorry like a million times.
Get on your knees if you have to and beg for forgiveness.
Do the whole damsel in distress act, and get all desperate.
Oh, and in the end, do the puppy dog pout just as a add in.
Say you are sorry, and sincerely mean it, if you do not mean it, or if you are going to do it again, do not apologize untill you are ready to stop...good luck..
Look them in the eyes, say seriously I'm sorry. Start crying and hug them.
That always works for me.