A very, very good blog post by author John Scalzi on apologies and apologizing. Please give this a read and make the world a better place.
First thing: What is an apology? Leaving aside classical definitions that are not directly on point to how the word is used in everyday life:
An apology is an admission that you’ve wronged others and that you are actually sorry for it. This is of course why it’s difficult for people to apologize. No one likes to admit they are wrong or that they screwed up. No one likes the complicated, defeated feelings that come with being wrong and screwing up. No one likes having to go to other people, publicly or privately, and admit to them they’ve been wrong and have screwed up. It is, literally, humiliating, since apologizing almost always requires humility and a willingness to put the needs of those you have wronged over your own. For ego-bound creatures, and we are all ego-bound, this is a hard thing to do.
I think it becomes less hard, however, if you consider the following:
One, everyone’s wrong at some point. Because, hello, you’re a human, and humans are imperfect beings. It’s okay to recognize you are not infallible.
Two, it’s better to center your ego on doing what’s right rather than never being wrong. Because, per point one, you’re going to be wrong at some point.