To me, an apology has three parts. They are as follows:
1. Say, "I'm sorry." I remember learning to say "I'm sorry" in elementary school. The students would have to apologize to whoever they did wrong. Then, the teacher would ask, "Now, what are you sorry for?" That's a tough question for an elementary school student. It's also a tough question for adults. The Institute for Healthcare Improvement (IHI) calls the "what I'm sorry for" part, acknowledgment. That means you have to admit that you did something wrong. In addition, IHI states that you should offer an explanation for what you did. "That happened because I engaged in this behavior and/or because I made a mistake." The actual "I'm sorry" portion of the response is what IHI calls remorse. That means, that you feel bad that you hurt or injured or caused someone else some kind of discomfort.
2. Promising to do better in the future. To me, an apology, simply offering remorse, is not enough. In order to demonstrate sincerity and offer a true apology, someone must promise to do better in the future. "I understand that my behavior/actions/mistakes were upsetting or caused you harm, so this is how I am going to do better from now on." This is what IHI terms reparation. In other words, you know you did something wrong, you learned from your mistakes, and you promise to make changes to adjust to make sure that those behaviors/actions/mistakes don't happen again.
3. Actually doing better in the future. This is the most important of the apology, as far as I'm concerned. Someone can apologize all they want, and they can promise to do better in the future. However, if they don't do better in the future, and they continue their same old behaviors, then what is the point of them apologizing in the first place? Of course, change takes time, and you can't force someone to change in exactly the time frame that you deem appropriate. You have to be realistic and understanding of the other person's wants and needs, as well. Change in a relationship, like conflict in a relationship, is not totally one-sided. If you are truly receptive to someone's apology, then you also have to be receptive and acknowledge their willingness to change for you. If you see no change but simply a continuation of the same behavior, or even worsening of that behavior, then you know the apology was insincere.