How do you apologize to your lover? The best apologies are those that are heartfelt, empathetic, without any backhandedness or fakeness. For example, if you have ever apologized by saying "I'm sorry, but..." that is an example of insincerity. Leave out the extra piece of your apologies and purely empathize. Do not try to win a battle with your apology or prove that you shouldn't need to be the one apologizing, for example. If you got into this situation, your partner just wants to feel understood, so empathize with her.

Dale Carnegie reiterated the previous point. It's what we most want as humans- to be understood. So instead of telling your girlfriend "I'm sorry that I didn't call you back, BUT I was really busy, and you don't call me back anyhow" or something along those lines, you should instead replace that with "I'm sorry for not calling you back, you must feel frustrated with me, I was consumed with a project that I had been working on..." etc.

So again - the big difference here is not arguing or trying to prove your point. You're not trying to win her approval. You are understanding HER point of view. We all just want to be heard, to be understood. We all have our insecurities, and once you understand that it will help you out tremendously in your relationships.

Apologies aren't always easy, and they change depending on the offending action. There is a lot of information on apologizing the correct way; it was included in the Magic of Making Up e-Book package when I purchased it a couple of years back.