We all make mistakes. If you are not making mistakes, you are not trying new things. When we make mistakes, and when those mistakes affect others, we make it a point to apologise.
Apologising is an immensely powerful thing for you to do, and contrary to what you've been taught, it has nothing to do with the other. A good apology is one where you take responsibility for yourself and your actions, and re-align yourself with the path that you've chosen. It is where you create personal power, by owning your missteps and changing your direction. It's where you evaluate and set standards for yourself and try your best to meet those standards.
It is often important to express an apology to the one(s) you've wronged, but not so that they can forgive you. An apology is not where you grovel, where you make yourself less, where you beg for them to stop being angry at you and to stop holding a grudge. It is not about getting them to drop their resentment and accept you back into their good graces. It is about communicating to them that you are aware of your own problematic actions and that you intend to change them. If they do not accept your apology, there is no problem, because it was never about them. It's about you.
There is only one time when you apologise, and that is when you feel that you have acted in a way that is not in line with who you want to be. For example, if you have decided to be a kind person, and you were harsh with your lover and said something critical when they were being clumsy, you were in the wrong. Not measured by some objective moral code, or by how upset they are with you, but because you failed to act in line with your decision. You acted in a way that you aren't proud of. You look at your own behaviour, and you wish you would have done it differently, because this isn't the person you want to be, and this isn't the behaviour you want to do. So you apologise. You communicate to yourself and to the other person that what you did, or said, or felt was unacceptable to you and you take steps to improve yourself.
A good apology looks like this:
1) "Here is how I acted." Be accurate in this, and describe as objectively as you can what it was you did that you're not proud of.
2) Optionally, "Here are the consequences of my actions." Communicate how what you did negatively affected the other person. This is optional, but goes a long way to help the other person feel that you understand them and care about them.
3) "What I should have done is ..." or "What I wish I would have done is ..."
4) "I apologise", or "I'm sorry for that." You're apologising not for how they felt, but for failling to live up to your own standards.
5) "I will try my best to do that in the future."
Some examples:
- "I was home two hours after I said I would be home. I can see that you must have been very worried. What I should have done is call you to let you know I would be late. I'm sorry I didn't do that, I'll call to tell you next time."
- "Last week you texted me a lot of times, and I ignored your texts because I was very busy. I thought about it for a while, and I can see now that you were just reaching out and trying to spend more time with me. I do like that you want to spend time together, and I don't want you to feel like my work matters more to me than you do. Now that you tell me how sad that made you, I wish I would have taken the time to talk to you. I'm sorry I ignored you. What I'll do next time, is at least let you know that it's work related and that I do care about you, and I'll tell you when I'll have time again."
- "When you told me yesterday that you had cheated on me, I screamed at you and said some very ugly things. I'm sorry for that, I'm not ok with saying things like that. I love you, and that is not how I want to respond. I want you to be able to tell me when you make a mistake. If I get angry and out of control when you are honest with me, you might not want to tell me next time something happens. I find it very difficult that you cheated, but what I should have done was to take some space to process, and then talk to you when I was calmer. Next time I start to get out of control, I will tell you I need some space, and get back to you later. I still want to talk about this cheating and where we go from here, can we talk about that tonight?"
A good apology shows clearly the difference between your actual behaviour and the behaviour you wish you would have done. It shows your awareness that your behaviour hurt the other, and it gives a very clear idea of what you intend to do in the future to avoid hurting the other person again in this way. This doesn't have to involve doing what the other person wants, or what they would like, at all. They may want you to be home on time every night, but if that is not something you can or want to commit to, then don't. The only thing you're apologising for is failing to live up to your own standards, not for how they cohose to respond.
After you've apologised, don't beg for their forgiveness, and don't expect anything from them. They have every right to feel whatever they choose in response to what you did. They can choose to be angry and stay angry, and while that may not be reasonable, you can't change that. They may still be sad. They may still trust you less than before. They may think that what you will do in the future still isn't good enough for them. (That's not your problem). They may say that there's no need to apologise because it wasn't a big deal to them. (That's also not your problem, you're apologising because it's a big deal to you). All of that is fine. Don't follow it up with "Now you also have to ..." or "Now that I said this, I expect from you that ..." . This isn't a trade. You didn't give them anything, you corrected your own course and took your own personal power back.
Lastly, a good apology needs to be followed by a change in behaviour. If you said you will do this other thing in the future, make it a point to do so.
A good apology doesn't just improve your relationships, it also - and more importantly - helps you to grow into the person you want to be.