Not all apologies are created equal, particularly when the offense for which one is apologizing is significant. The best apologies are those that both signify regret and also help the hurt partner begin to heal, and they have some very specific characteristics.
Many apologies come with an explanation or caveat. "I really didn't mean to hurt you when I wrote those inappropriate emails to my ex-girlfriend...I was just feeling so lonely!" a husband might say. This type of apology implies that his wife was somehow responsible for his bad behavior. But she's not. She may not be paying enough attention to him, and for that she is responsible, but he has a host of possible responses to that problem. He could tell her he feels lonely, request that they see a marriage therapist, or write that email. He chose the email option. It's his responsibility.
And now she's hurt. The best type of apology in this case would be something that acknowledges his full responsibility for the action chosen, plus suggests a way to make the future look better. "I didn't mean to hurt you and promise not to hurt you in this way again. I'm deleting the person from my email files, Facebook account and Twitter list today. If she contacts me, I promise to let you know. Is that a good start? And may I hold you and tell you I love you?" In her hurt, she may still lash out at him, but his response acknowledges both his responsibility and her pain...and provides the foundation for positive forward movement.
And I want you to notice that in no place in this apology does he suggest to her what she might do to make things better - he focuses only on his own actions.
In cases of significant trauma that impacts trust, such as an affair, really big lie, or long-term issues around reliability such as are often seen when one partner has undiagnosed ADHD, a single apology is not enough. A long-standing 'posture' of apology is often necessary for an injured partner to recover. This is sometimes hard for the "offender" to sustain. "I've apologized...why can't you just get over it and move on?!" is the often angrily asked question. While a single, or even string, of apologies is helpful, what rebuilds the damaged trust is the actions that accompany the apology. Is the husband who wrote the emails willing to listen to (and validate) his wife's pain over the long-term and put his arms around her when she cries, even a month later? Or was he just apologizing because it was the most expedient way to get out of the hot water he found himself in? Does he follow through on his promise to have no contact with the ex-girlfriend over time, or does she remain somewhere in his life? His longer-term 'posture' of apology is the series of actions and continued validation of his wife that proves his dedication (and apology) are real.
And, again, adding an explanation or caveat doesn't work well.
This is not to say that the husband shouldn't seek to solve the underlying problem - his loneliness. But unhooking it from his emails means that the couple can focus on the underlying issue, not on his bad choice. He can continue to take responsibility for, apologize for, his bad decision while also asking that his needs be addressed in a meaningful way. (And it's almost inevitable that once he opens up about his own needs, his wife will start talking about hers...)
Next time you need to apologize for something that was a breach of trust, make it straightforward and without caveat. Suggest a way you yourself can correct the problem. Validate your partner's hurt over the long haul. Seek to address your underlying issues in a constructive way. Listen to how your partner responds. You'll find it's the best kind of apology.