How to Apologize Sincerely When You Realize That You Have Hurt Someone

1). Step 0 is to recognize that you've done something wrong. Maybe you didn't, but there's a good chance that you did. If you're still at the point where you think that whatever you did is "not a big deal," or that the other person is overreacting, or that the other person does things to you that are far worse than what you do to them, don't move forward until you are past that stage.
2). The next step is to fully understand why you are apologizing. It's important to apologize for the right reason. If your wife told you to sleep on the couch because you did something that upset her, don't just blurt out, "Sorry, OK?, Damn!," with an "attitude" while rolling your eyes. That isn't an apology. Sometimes you may immediately realize when an apology is in order, and sometimes it could take a long time. Sometimes it could take many years, and then the day will finally come that you fully understand what you have done.
3). The next step is to decide if you are truly ready to apologize. Perhaps you fully understand that you've made an ass of yourself in some way, but you shouldn't apologize until you can do so sincerely. Don't just apologize because of the awkwardness of having the other person staying mad at you. Don't apologize because they are withholding companionship or anything else from you. Don't apologize because you are sorry that you "got caught." Wait until you are truly ready to apologize for its own sake, and for the benefit of the other person.
4). Before apologizing, you need to have a true understanding of how you made the other person feel. This is known as empathy. You need to put yourself in their position, and imagine being on the receiving end of whatever you've done. Keep in mind that even if you are the type of person who thrives on being abused, it doesn't mean that the offended person is the same. It's not sufficient to say, "Well I wouldn't have gotten upset if she did the same thing to me!" All that matters is how you made the other person feel.
5). When you are ready to apologize, don't just say, "I'm sorry," and leave it at that. Explain why are you are apologizing. Prove to the other person that you understand what you did. It could be as simple as, "I'm sorry that I didn't take out the garbage even though you nicely asked me to several times," or something broader like, "I'm sorry that I'm such a bad listener, and I always cut you off before you are finished talking." With that said, don't apologize sarcastically in a meaningless, generalized way such as, "OK, I'm sorry that I'm so imperfect, and can never live up to your expectations, and I'm sorry for being born, and for being the worst person on Earth." Obviously that will just make things worse.
6). In addition to apologizing and proving that you understand the situation, acknowledge your understanding of how you made the other person feel. You might say something like, "I know how angry you must have been seeing me flirt with all the other women at the party," or, "I know that you worked so hard to prepare dinner, and you must have felt really hurt when I said that the food was bland."
7). Consider explaining to the other person that you will try to not repeat your transgression in the future, but don't make promises that you can't keep, or that aren't realistic. If you got busted for fooling around with another girl, don't say, "I'm sorry, I'll never talk to another woman ever again!" No one is so stupid to believe that, nor is that even the point, nor is that even necessary. However, you could say, "I'm going to think about why I felt the need to do what I did, and I'm willing to go for counseling and to do whatever is necessary to preserve our relationship." Conversely, don't say something asinine like, "Honey, why are you so upset? It's not like she was hot or anything, and I don't love her like I love you."
8). After you have apologized, you simply have to be brave and accept whatever comes. Your apology might be accepted wholeheartedly. You might be accused of being full of crap. You might get smacked hard across your face. You might get spat on. Whatever happens, happens. This is your destiny working itself out. You're reaping what you sowed. You did the right thing by apologizing, and if you did so sincerely, then in theory everything will be fine. If not, you have to accept that as well, but at least you did the mature thing by apologizing. That is infinitely better than not having done so.
9). Apologizing can be extremely difficult, but it gets easier with practice. Eventually you can get to the point where you immediately recognize when you've hurt someone, and you can apologize quickly and sincerely.