Relationships often end messily, with emotions running high and many things left unsaid. Everybody deals with a breakup in their own way. Some people throw themselves into another relationship to help them get over the pain. If your ex-girlfriend has already found herself another guy, it may have made you realize that you are still in love with her and want her back. It's possible that you could end up back together, but this is a time for patience, not recklessness.
Accept that your relationship ended for a reason. Spend some time thinking about the reason and trying to learn from the experience. No matter who ended it or who was most at fault, it's highly likely that both parties made mistakes and could have dealt with issues in a healthier, more productive way. By admitting to yourself things you may have done wrong and working out how you can avoid making the same mistakes in the future, you will be in a far better place to reconcile with your ex.
Take it slowly. If you are not in touch with your ex-girlfriend, send her a text or email to re-establish contact. If you treated her badly, apologize for your actions. Don't pressure her into meeting up or bombard her with questions at this stage. Give her some space and time to think about the fact that you have reached out to her.
Show your ex-girlfriend that you respect her. Mutual respect is one of the key elements of a solid relationship. Let your ex-girlfriend know you are there for her if she needs you. Stay in regular contact with text messages and emails. Spend time with other friends and keep up your hobbies. If you find yourself with spare time on your hands because you no longer have a girlfriend, take up a new interest. She won't find you very attractive if you are sitting at home all the time pining over her. Get on with your life, for your own sake.
Be honest with your ex-girlfriend. Tell her you still have feelings for her and would like to give the relationship another go. Resist the temptation to badmouth her new partner, whatever you think of him. Tell your ex-girlfriend you have learned valuable lessons from your breakup and that you consider yourself to be a better person and the right guy for her. Give her time to respond. Don't force her to make a decision before she is ready. What happens next is up to her.