Group rights are extra privileges that a user can be given. All users can create and edit pages, but some users have the power to protect, delete or revert pages. These powers are taken very seriously and all users may register any complain they have with the Appeals Group if they feel these responsibilities have been abused.
If you have been granted these extra permissions, this means you will have been inducted into the corresponding group. This accolade is not automatic, but is overseen by the Special Privileges Council.