To become a moderator, you need to simply prove you're a good member around the board, and are loyal (by being here for a long time could help that). Also that you don't bring arguments to the board, or participate in them. If you try to be a peace maker too between people fighting, that's even better.
The point is, you don't apply to be one. You're just let in when you are seen as someone who would make a good one.
One thing you wanna avoid is asking for it or begging, but asking this first one time doesn't really hurt..
Just have fun on the board, be a good guy, and if you seem worthy, you'll become one.