You've done something to make me upset. You bastard. You'll say that you're sorry because you're trained to do that, and that's good, because if you don't, then I'll be really angry at you, but other than avoiding those negative one thousand points, the apology doesn't mean that much to me. I've been the recipient of too many false apologies that the transgressor didn't mean at all, seen the same happen to others around me, and seen even more of it happen in TV and movies. It is just a thing you said when you noticed that I was upset. Do you even know why you said it? It's worth bonus points if you can tell me what you did wrong in your apology. But still, an apology is not what I'm looking for.
What I'm looking for is a correction. A call to arms, a plan of action; something you're going to do to make things better. I don't want to be upset again. Tell me how you're going to make sure that you don't upset me again. Let's say you forgot to take out the trash. This was your responsibility and you've forgotten for several weeks now. I came home to find that the trash can outside was knocked over and spilled all over, and the trash inside wasn't ever even taken out. Now I'm angry.
Bad answer: (make no indication that you even notice that I'm upset)
Passable answer: Oh, I'm sorry.
Okay answer: Oh, I'm really sorry that I forgot to take out the trash again today.
Correct answer: Oh, I'm really sorry that I forgot to take out the trash again today. Next week I will set everything out the night before so I don't miss it again.
That "basic" apology without a plan of action pretty much just works on me once. By the second week it's already meaningless, serving no purpose other than to acknowledge my annoyance. I'll probably roll my eyes or sigh when I hear it the second time. Telling me that you're sorry is one thing, but telling me how you're going to correct the problem shows me that you're sorry. That's what's meaningful.
Next time you aggravate someone, spend an extra moment to think of how you can either fix the problem that you've caused, or at least make sure that it's less likely to happen again. Then tell them what you've come up with. I bet they'll appreciate it.