The dynamics of repairing a relationship change depending on which side of the relationship it is. For the woman trying to get the man back, it's less about what you say and more about getting him to tell you things, and you really, truly listening to him. Hoping for an ex boyfriend back can be the easiest or the most impossible thing, depending on whether he's moved on or not.
If he left you for someone else, you probably know from experience that that's barely even an impediment. Unless they're picking out china patterns together, the door's still open at least a crack for you. He may decide that new horse isn't as much fun to ride as it looked. She may have been playing him the whole time and dump him, and now you get to play pick-up-the-pieces. He may suddenly snap to the realization that you really were the one the whole time, and come back for you. Men tend to do this out of the blue, like when they're out jogging or in the middle of completing a Call of Duty IV map or something.
If he just plain left you because of you, that's a different story. If it was physical (it's surprising how seldom it is), it's easy to see that hitting the gym or getting a make-over will at least help. But more often, it was over personality issues. That's a different story! Hoping for an ex boyfriend back becomes a huge struggle if there was something about you that drove him away.
If you have past issues, such as baggage, emotional scars, past abuse, or other trauma, it's time to get your head together. You may need to see a doctor first and a boyfriend second. Problems such as depression, anxiety, panic attacks, post-traumatic stress disorder, bipolar disorder, and other mental issues are a tough handicap in relationships. It's difficult enough for men to understand regular women, without having extra complications there. Your best bet here is to acknowledge that you have a problem in this area and point out to him that you're willing to work with him on it. Some men just want to help you feel better.
Another common scenario is where you let your anxieties about past relationships get in the way of this one. If so, then watch out: you may be developing issues towards all men, not just this one. Once again, admitting to these problems and even offering to go to counseling together will go a long way towards easing those issues.