As @ProfSrlojohn said, there is no formal application process. The moderators are hand picked by the staff members whenever new mods are needed.
In Slime's case, he had sent a message to TTV asking to become a mod. And after some deliberation by the staff, he was made one when new mods were needed. However, Slime is very much the exception to the rule. Generally staff members are the ones who send messages to people they want to hire.
As such they compose a list of potential mods. So if you want to be on that list, you'd better be an upstanding member of the community- flagging spam and so forth and who knows, whenever new mods are needed they might consider you.
There are many different requirements involved with maintaining Master. You can always PM a mod to find out which requirements you currently aren't meeting.