I have three ways to apologize to your girlfriend. They have worked for me and others that I have shared these ideas with. They don't all need to be used at one time. These ideas are "stand alone" ideas. They can be combined, but it is not necessary to combine them to bring forth a meaningful apology to your girlfriend.
The first step, and important step, is to actually apologize. Women or girlfriends want to hear the actual words. They want to hear, "I am sorry." Or they want to hear, "I really do apologize." It needs to be said in a manner that portrays honesty. No matter what brought on the apology, you can adapt your apology to the situation. Some examples are as simple as, "I do apologize that I let this situation go on without resolution." Or you can also say, "I am sorry that we disagreed and it escalated into a fight with you." You cannot apologize to your girlfriend if you aren't feeling the apology. By that, I mean that it has to come from you with an honest effort to begin the healing process between the two of you.
Another way to apologize to your girlfriend is gifting. The one thing you want to remember is not to "over" gift. Most girlfriends would enjoy a gift, but if you give a really big gift as an apology then you will be expected to buy more expensive gifts for special occasions. You can gift a favorite CD or DVD movie that perhaps you both can watch or listen to as you begin to recover from the situation that lead to your apology. You can give her a book or maybe a diary for her thoughts every day. Dig deep in your memories with her and try to recall anything that your girlfriend would have mentioned that she wanted someday. That would be the one thing that you would want to gift. She might have even mentioned having the faucet fixed or the backyard mowed. Do those as gifts from your heart to show that you meant no harm and want to apologize with some sweat. Or you could make up a homemade coupon book and let her redeem them when she wants to redeem them. They could be all the things that make her happy when you two are together.
Try to remember a place where the two of you were together and happy. A place that you two have shared before. Take your girlfriend to that place to share some peace and quiet together as an apology to your girlfriend. This lets your girlfriend know that you are attempting an apology by remembering good times with her and wanting to share those memories. A lot of times memories that were made during really quiet times or peaceful times serve us later in our attempts to bring a person out of an angry state with us. That in turn lets them know that you are apologizing for whatever the situation was that caused the need to apologize. It isn't a fun thing to do so going somewhere that no one can bother you two would allow your girlfriend to know that your attention is solely on her and apologizing to her. It could even be a favorite set of stairs that you two used to sit in while getting to know each other. It could be the library where you two used to study together. You can even apologize in the parking lot of a busy grocery store. Just let it be a place that was you and your girlfriend's place together.