From what I've heard the Admin/Moderator staff scan through the active members that have been on the site for at least a year and observe those who they think would be a good potential Mod and when a position opens they offer it to them.
To my knowledge there is not a public application process.
What you can do to help your chance of being selected is to be active on the site, answer questions and give good solid advice (over and over again as many common questions are asked frequently). Be friendly and welcome new members to the site. Also you may want get to know the current Admins & Mods (there is alot of trust involved, they are less likely to offer the job to someone they are not very familiar with).
I don't know for sure if there is any pay or incentives involved. If I had to guess though I wouldn't think so.
For more information I would recommend messaging JLC or another member of the Admin/Moderator staff.