To me, the most important part is that you mean it. I personally don't usually accept "apologies" because anyone can apologize... but it's a matter of semantics with me.

Asking for forgiveness that's what impresses me. State what you did that you are uncomfortable about, that you shouldn't have or didn't mean to (whichever is true) and then ask them to forgive you. As a closer you can say something about working "on that" so it doesn't occur again, etc.

By stating what it was, the other will know you know what you did that offended them. Of course, too many people today think they have the right to NOT be offended. Make sure an apology is in order, and even if it's just for your own health, give it.