There is no "process" or application ... in fact asking to be a moderator will get you on the 'don't ever ask that member to be a moderator' list
The only "criteria" is just be a member in good standing ... one without any warnings and/or infractions, or temp bans.
All staff members were originally hand picked by the forum owner, Frederik Magle when he felt the need to add moderators. I was chosen in 2007, after having been asked to be on staff at our sister forum, MIMF, the year before, I was made Sr. Moderator, then Assistant Administrator in 2008, and Administrator in 2011.
When the time comes that we feel additional staff are needed, we come up with a list of names to consider. After much internal discussion we finalize that list, and then we will ask that/those member(s) if they are at all interested in joining the staff.
If they don't accept that is fine ... there will never be any problem with decliniing nor would we ever think differently about that member in the future.