In my office, over a bowl of mint chocolate chip ice cream cake, a coworker and I had a rambling conversation, at times pragmatic and at times emotional. The topic: can someone overcome a major, offensive faux-pas? (Think: Don Imus.)
I started off by saying that Imus hit a variant of the "male apology."* The words "male apology" were barely out of my mouth when my (male) coworker said "Ah yes - 'I'm sorry you're upset'!" It unnerved me that he knew exactly what I was talking about without any elaboration. The variations I was thinking of were "I'm sorry I got caught" (aka "the cheater's version") or "I'm sorry this is negatively affecting me" (which fits Imus to a T.) All variations take no responsibility for the speaker's role in the situation. More than that, they express little or no interest in learning how or why the speaker's actions have resulted in this situation.
So, what am I looking for in a sincere apology? First, I'm looking for the apologizer to take responsibility for their part. I'm not saying they have to take all the blame (My mother's passive-aggressive tactic which makes me see red! "Fine, fine - it's all my fault!") Also, it's not enough to claim to take responsibility without manifesting an understanding of your role. ("I own up to my mistakes.") And, second, some basic contrition - some understanding that your words, actions, failures to act caused someone else pain, or unhappiness, or a long, cold, wait in the rain. Whatever.
It's not so hard really: take responsibility, say you're sorry and mean it.
* - It's called the "male apology" because guys use it more often. BUT, I'm not averse to using it where circumstances require.