It’s actually one of the hardest parts of being human. Acknowledging the fact that you could be wrong, and that your actions hurt another person. Or that you are right, but you still hurt another person. Ultimately, the question is, what does a descent person do in these situations? The answer is, apologize.
Now what exactly is an apology? Is it an admission that your actions were wrong? Is it an admission that your actions caused another person pain? Is it an admission that you went outside the social contract and did something that was pretty nasty? Is it an admission that you were thoughtless, and uncaring? Is it an acknowledgement that how you made someone else feel is more important than being right?
Maybe yes to all, maybe no. An apology can be simply the result of the fact that you, on your own, reviewed your behavior and found it wanting in some way. It is the mindful way of looking at your life to acknowledge that you are not perfect, and though even if you meant no harm, there was harm involved. That even if you had the right to lash out at another human being, was it really worth it in the end? Was it worth it to you to cause the hurt and the harm? Did you really get your point across, or did you just make yourself out to be an unfeeling, rather self-involved person?
An interesting religious point is that under Jewish law, a person is obliged to accept an apology. For the apology is not about you, but about the individual recognizing that they did wrong. It is their reaching out for forgiveness, to fix their soul. In fact, if you are loathe to forgive them, but they try to apologize three times, then you are obligated to accept, as their continued attempts at atonement proves that they are truly sorry. However, without a doubt, you could assume that in some cases that would still be too much of a burden to bear. For all hurts are not equal.
Say if someone belittles you in public, but later comes to regret what they did, and they apologize. Then by all means accept the apology. Of course, it is also Jewish law that under these circumstances, a person who insults another in public has to make their apology public as well. They are to be held up to derision in much the same way that they caused the hurt they created. In other words, the apology must carry the same weight as the insult as far as public admonishment.
But in truth, if you apologize in public then that would also cause the public to think better of the perpetrator, because they acknowledged their own wrong doing. So in some sense, this is a win for both parties. The aggrieved is made to feel better, as their hurt is acknowledged, and the perpetrator, while acknowledging their own wrongdoing, is proving to the public that they are able to uphold the order of society, and that they should not be considered charem, or outside the community.
To prove to those you live amongst that you are able to live a fruitful, productive, and positive life, that you are able to add to the society in which you live, is an important part of the human story. While we never really like to talk about it, there are rights and wrongs for behavior that have nothing to do with law and punishment, but have everything to do with how a society functions for the betterment of all involved. The kinder the people within your sphere, the better lifestyle for all those concerned.
On the other hand, what if someone murdered a loved one? Are you required under law to accept the apology? Here is the interesting Jewish answer: only the one harmed can accept an apology. Hence, technically, you can never be forgiven for murder, because the murdered person cannot speak from the grave. It is an old tradition that yes, family can forgive if they want to. But it is by no means a requirement to accept such apology even if given three times. To murder is the most heinous of crimes, and a crime that is not easily forgiven, if ever. Understand, Christian law on forgiveness is different, as is Islamic law.
Also, it is important to remember that under Jewish law, in order to atone (seek forgiveness, or apologize) for crimes against society then you must accept society’s punishment. God cannot absolve you of wrongdoings to your fellow human being, only the community can absolve you. When you read about Jews fasting on Yom Kippur, hoping for atonement, that is so that they are forgiven for their sins against God. The day has nothing to do with any hurt you have caused toward another person, even if many persons take that opportunity to apologize for any harm caused to another.
The truth of the matter is that to apologize is an important part of being able to think outside yourself. To be able to say that there is something more important than being right is to acknowledge that even if your facts are correct, your mission as a human being is ultimately to make this world a better place, reduce hurt, harm and mean spiritedness. A goal for leading a good life is simply to do what is right, and sometimes that means you need to apologize, even if you are not wrong.