How to Apologize to a Girl

 1 Acknowledge apologizing as a way to move forward. No matter why you are apologizing, remember that the goal is to make up with the girl and improve your relationship. At its core, an apology is an emotional concession that can have great returns if properly and sincerely delivered.   

• It has been found consistently in studies that men have a far more  difficult time of apologizing than women. If you count yourself among  those who find difficulty in apologizing, approaching the act as a  purely practical manoeuvre (at least at first) can help you get over  that first hump.

 2 Give yourself time to relax.  If you need to make an apology to a girl, then you are probably dealing  with some pain as well. If you recognize that you're not feeling good  at the moment, you should see first things first and give yourself time  to heal. Depending on how bad the situation is, this could mean minutes,  hours or days.   

• Don't let yourself put the apology off too long, however. Being  silent long enough will be interpreted as a non-apology; that you aren't  sorry and don't want to have anything to do with her. Again,  interpretation of what is considered 'too long' will depend on the  severity of the situation, and the strength of the relationship.

 3 Understand why she is upset. Apologizing without  meaning it or knowing why can make things worse than if you simply kept  your mouth shut. If you rush into an apology,, the woman will detect  you're not being honest. Before you apologize, it's important you take a  moment to stop and reflect. Why is she upset? Are the possibly other  factors that are playing into her mood? How serious is the problem?  

• Put yourself in the woman's shoes for a moment. While you should be  considering the way she is feeling, you should also imagine the way your  actions have been interpreted by her in turn. If your apology is  because of a certain thing that happened, try playing it out in your  head. From there, regardless of who is to blame, you should have a  better grasp why she is upset.
• Keep in mind that empathizing with why she is upset is distinct from  necessarily admitting guilt. Even if you don't think you have done  anything wrong per se, a healthy relationship involves recognizing where  the other person is coming from. Even if you think at first her reasons  for feeling this way are unjustified or even irrational, it's important  to recognize that the pain is real.

 4 Be sincere with your apology. If the problem in  question is severe enough, a woman will be analyzing your delivery for  any possible signs of fakeness. The best way to ensure your apology is  received as you want it is to make sure you actually mean it. If you  think you'll need to grit your teeth as you apologize, it's a good idea  to hold off and reflect upon the apology some more, or do without one  altogether.   

• You may be very angry for the same reasons as the girl. This will  make it hard to open yourself up to her. If this is the case, give  yourself some more time to relax and breathe.
  

(source: http://www.wikihow.com/Apologize-to-a-Girl)