So, you've been dumped. Sorry to hear that, really. Hopefully you can take some small comfort in knowing that you are not alone in that regard. There is a lot of good advice out there, but the catch is that you actually have to take action. And you should do something to win your ex back if you still have feelings for them. But that doesn't mean you should play head games to do it.
If you have only recently broken up, then the first thing you should do is cut off all contact with your ex. You both need some time and space to be alone so you can sort things out. That means no emails, no text messages, no phone calls and no tracking them down so you can talk to them in person. If your ex is the one constantly trying to contact you, then be firm and tell them you need some alone time.
On the other hand, if the two of you have been broken up for a while, and haven't had any contact, then you need to get in touch with them again. This can be tricky. You don't want to lay it on so thick that you only push them further away. Instead, use a low key approach–especially at first–and be polite. The purpose of your first chat is to reopen the lines of communication; to get your ex used to the idea of hearing from you again.
Now, the longer you have been apart, the higher the chance that your ex will be seeing somebody else. As hard as it may be, you need to respect their right to be with someone else. Also, do not hold any grudge against the new person in their lives. But even if they are with someone new, don't give up hope. You can still try to contact them once in a while, and then all you can do is bide your time and see if they break up. That's when you can make your move.
But why would they get back together with you, when the two of you have already broken up? That's a fair question, but one without an easy answer. The truth is that it can take a lot of work. You need to dig below the surface and get to the root of what led to your break up. For example, you may think that too much arguing was the cause of the split, but arguing is only an external symptom of a deeper problem. Your task is to find what that deeper problem is.
Once you have that worked out you have a few options. You can fix it, ignore it or forgive it. If it's something that could be a problem if the two of you get back together, then you need to fix it. If it's a minor issue, and one you are sure won't be a problem, then you may choose to ignore it. If it's something that hurt you emotionally and can't be undone or fixed, then forgiveness is your best option.