If you've figured out what you're actually sorry for, you're ready for step two of a meaningful apology: 2. Explain your state of mind.
An apology is as much an explanation as it is an expression of regret. If you're sorry for your behavior and the way it made someone feel, explaining your state of mind really helps the apology gain some traction. One of the most difficult parts of forgiveness for the person who feels wronged is understanding the mechanics of how and why you did what you did. What were you thinking? What were your motivations? Did you not realize how hurtful this would be? The person you’re apologizing to is definitely asking themselves these questions, so answer them preemptively during your apology to show you've really thought this through.
Don't simply say, "I wasn't thinking, I'm so sorry." Take accountability.
If you were acting selfishly, now is the time to fully acknowledge that. For example, if you cheated, it's infuriating and deeply frustrating for your boyfriend, girlfriend, husband or wife to hear that you just 'weren't thinking.' Own the fact that you made a conscious choice and that you risked your relationship:
"I lost perspective and stepped out of my character in a way that I can't take back. I risked our relationship, your trust and respect in me, all the memories we've built together. I risked all of that and made a decision that was so deeply selfish. What lead to me losing perspective is how little I've been taking care of myself, how consuming X has been, how anxious I was about Y and how I just avoided the anxiety instead of dealing with it like an adult. I see now that it's always when I don't take care of myself and my life that I make these incredibly destructive decisions, and I am determined to re-prioritize staying on top of my life to ensure that I remain in a healthy place. Nothing is worth the risk of losing you and when I'm addressing what I feel instead of just avoiding it through self-destructive stupidity, I can obviously see that clear as day."
The above is a significantly more meaningful explanation than, "I just wasn't thinking." Take accountability, acknowledge your lapse in judgement, try to explain the conditions under which the lapse in judgement was made, then explain the reasons that lapse in judgement will not happen again.
Communicating self-awareness about your state of mind during the time that you committed the offensive behavior is absolutely crucial. Showing you have self-awareness about what you were thinking and feeling assures the person who is receiving the apology that you were not in the same state of mind that you're currently in, and therefore won't be repeating the offense. Demonstrating increased self-awareness is also reassuring to the receiver of the apology because it implies that should you slip into a similar frame of mind in the future, you'll be able to immediately identify it (because you're so self-aware) and with similar immediacy, you'll be able to choose a different course of action.
Remember, an apology is a process and not an event, just like forgiveness. A meaningful apology takes time to express and time to absorb. There are four whole other steps, so patience with yourself and with the receiver of the apology goes a long way.