I recently read an article from a BYU Devotional. It's called, "Repentance and Forgiveness in Marriage," by Rick Miller. Don't let the title throw you. It should actually be called, "How to Say I'm Sorry" and is valid for every person and every relationship.
Dr. Miller boils it down to a few points. 1) Say what you're sorry about in complete sentences. 2) Admit you were wrong, offensive, out-of-line, etc. 3) Don't include any kind of excuse or justification. 4) Repeat as often as needed, never demanding forgiveness--remember, you were the one who made the mistake.
As I see it, there are three different ways to approach another person when you've made a mistake.
1. The Non-apology Apology.
This is characterized by saying something like, "I'm so sorry you were offended." It sounds like an apology because it does include the words "I'm sorry," but those who use this method aren't fooling anyone. In this apology, not only does the wrong-doer exempt themselves from all responsibility, but it also foists all responsibility to the offended party. In this way, the one giving offense now places themselves in the role of the victim of the person who took offense. Effectively they are saying, "Here we were having this conversation and now you're offended and I didn't even do anything! How rude of you to get offended!" This type of person lacks introspection and is fixed on being right rather than making things right.
2. The Justification Apology
This is the ever present, "I'm sorry, but..." This is an actual apology, or at least a good attempt at one. These people can at least see that they did something wrong and have the courage to admit it. But they also see WHY they did things wrong and want the excuses that comfort them to somehow comfort those they have wronged. They may go on to explain that they were stressed, tired, hungry, sick, etc. Something is added that softens their level of responsibility. They admit they made a mistake, but they were not completely responsible for their actions because of some extenuating circumstance.
This is probably the type of apology most of us make, yet we don't really like this either. As the offended party, we often react to this kind of apology thinking, "I don't care how (stressed, tired, hungry, or sick) you were. That doesn't excuse your behavior." And they're right. It doesn't.
This brings us to our third and most ideal apology.
3. The Real Humble Apology
In this apology we take full responsibility for our actions. We state what we did wrong in complete sentences. In Dr. Miller's talk he uses, "I'm sorry that I didn't do the dishes last night like I agreed to." Just a quick, "Hey, I need to say sorry" isn't enough. The person should explain what they did without re-harming the individual. You don't need to say, "I'm sorry for calling you a jerk." But you could say, "I realized that I was out-of-line in my comments yesterday. I'm sorry for the offensive things I said." It is important that you don't interject any form a justification, excuse, reasoning, etc.
This kind of apology takes a lot of humility and a lot of courage. You walk out on a limb and say that you were wrong and you're really sorry--then you stand there. You're done. It's actually much quicker to apologize this way! But you also have to finish. You don't know how the person will react. They might be deeply hurt or still angry. And you are left with no excuses or blame to hang on to.
In an ideal world, they'll frankly forgive you. But sometimes, when people are deeply hurt, they may lash out. They may say things like, "You always do that. You'll never change. Why are you so rude?" or a myriad of other things. If you start blaming them, you've just switched from a real apology to a fake one. If you start justifying your behavior, then it's turned into apology number 2. It's incredibly hard to remain humble in the face of a counterattack, but if you want to complete the Real Humble Apology, remaining humble is what you must do.
To give this kind of apology takes a high level of introspection and a good deal of practice. As you try to apologize in a real way, you will notice excuses creeping in. Practice your apology before you give it. Write it down if you need to. Look at it critically for excuses or justification and then take those out. Prep yourself for their possible bad reaction. Commit to taking whatever they dish. This is sometimes called sweeping your side of the street. If they want their side of the street cluttered and dirty, that's their prerogative. Don't try and correct them. Focus on your side of the street only. And always pray for Divine assistance to help you as you apologize. If you are in tune with the Spirit, your utterance will be stopped if you start to blame or justify and you will be fortified to withstand any reaction.
The reality is that repentance is very sweet. And it's something that each of us needs on a daily basis. I feel most sorry for those who dish out only non-apologies, hoping that if they bring to God a life free of repentance, that He will be overjoyed that they never needed or used His Son's Atonement. I think they will be sad when they learn that the Atonement is a gift to be used not to be kept in a box somewhere until resurrection. It will only prove that they never truly understood the Atonement or all that it could do for them. For me, I want the gift of Atonement used over and over again in my life. Every day.
One of the best things about the Atonement is that it is a gift made of love, so no matter how much you use it, it never wears out. In fact, this individual Gift become more beautiful with use.