Friendly/Personal Apology Letter
If this is a personal letter you should start the letter by saying that you are sorry to the recipient. Next you should admit your fault and take responsibility for your actions. Next you should volunteer or ask if there is any way that you can help out to resolve the situation. Then you should let the recipient that you will try to make sure that the situation will not happen again. To close off the letter, you should apologize again. When writing a personal apology letter it should come from the heart and be sincere.
Formal/Business Apology Letter
If this is a business letter you should start the letter by saying that you are sorry to the recipient. Next you should give an explanation as to what went wrong. Then you should try to rectify the problem. To close off the letter you should apologize again.