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To understand what a publisher can do, we need to clarify the Access Control Levels (ACLs). The ACLs set the permissions that are associated with the individual User levels. There are seven User Groups/levels within Joomla only four of which have access to the website Frontend: Registered, Author, Editor and Publisher. There are also three levels with access to the Administrator Backend: Manager, Administrator, and Super Users. In addition to the registered User Groups listed above, there is a Public unregistered or guest, level.
A complete explanation of Permissions is shown at the Access Control List Tutorial.
A publisher is a User level with access only to the website Frontend functions, but with permission to publish content on the Site. You can examine the Publisher permissions in the Administrator: System → Global Configuration → Permissions tab. Then select the Publisher side tab.