Mod, Admin, and Staff are all very important built-in groups in discourse and they have many similarities.
Moderator
Moderators can moderate all content on the forums. This includes all the permission that all trust levels have. They also can delete all posts, access the flag queue, and bypass all limits including liking and editing. In addition to moderating content they can also moderate users. This includes
- Suspending
- Silencing
- Changing a Username
- Changing Trust level
- Changing the profile picture
- Changing the card background
- Changing their profile header
- Changing their about
- Anonymize users.
- And other small things
Moderators can do this on any non-staff user.
Admin
Admins have all the power. In addition to moderator perms, they bypass everything even category restrictions. They can also change the settings of the forums as well as add and/or remove moderators and admins.
Staff
Staff is people that are moderators and/or admins.