This is a very delicate situation and some feelings might be hurt so be weary about what you are doing exactly.

[edit] Steps

Find out why she broke up with you. What did you do to make her dump you? Girls don't dump a guy out of the blue unless they have a very good reason. Find out what went wrong, could it have been prevented, and how it can be fixed.
Apologize. If it was entirely your fault let her know what you did was wrong and make sure she knows you're sorry.
Continue socializing with other people. Just because you and your ex are out of contact temporarily, that doesn't mean that you need to be cut-off from the rest of the world. Date another girl if you want to. Causing some jealousy in your ex partner can work to your advantage if you're trying to get them back.
Let your ex know you have "moved on", but let her know you still care about her. That is how you get her back. She might think you moved on completely unless you say that. This will make you seem more mature.
Become friends with your ex again. This builds trust and will make you closer to each other. Afterwards, ask her why you and her broke up exactly. You will get one of the two answers: she will either think it was a mistake that you two broke up or she will think its for the better. Either way still stay friends with her.

[edit] Tips

Don't seem desperate.
Ask yourself if you will be able to accept possible rejection from your girlfriend again.
If all else fails, move on.
For additional help, read this

Determine why you broke up in the first place and make sure that you don't make the same mistakes.
If you start casually dating someone else, consider how this girl would feel if she knew you were using her to get back at your ex or make her jealous. Probably not good. Make sure you're also okay with that.