Manual:$wgGroupPermissions

Details
$wgGroupPermissions is a two-dimensional array indexed by user group and available permissions; for example, " " allows registered users to edit. These groups can be assigned to users through the wiki Special:Userrights interface.

For in-depth documentation, see Help:User rights.

Extensions
Extensions such as RenameUser or MakeBot add new rights which can be configured and assigned in the same manner.