A moderator on the Art of Problem Solving website is a non-administrator user who has powers for a particular forum which non-moderators don't. That is: they can lock topics in a forum, move them, delete them, comment about certain posts and so on. The short form for moderator is mod, the short term for administrator is admin.
When a user is an Administrator, their name shows up in red. When they are a mod, their name shows up in green.
An user automatically becomes an administrator of a forum if they create it. This user can manage who gets to be a mod or admin in said forum.
Moderators vs. Administrators
Administrators are more powerful in Art of Problem Solving than moderators. Moderators answer to Administrators, and not vice versa. Administrators can edit who is a mod or admin, but moderators cannot.
Some admins might have their names in blue instead of red. Those are AoPS site administrators. They are administrators to all forums on AoPS, even private forums and class forums. They are part of the AoPS ninja squad, but all work for Gmaas.
List of Admins
Below is a list of some of the current AoPS site admins:
5space (Dustin Ransom) aarjunakani (Arun Arjunakani)
There's way more!