The Staff Override permission allows the user to edit certain fields, (such as the Recommended Amount and Approved Amount) for Request records that have been assigned to them. It also allows them to edit their locked Requests (and any attached, locked Documents) without first unlocking them.

This permission is an "override" because the user is allowed to edit these fields for her own Requests even if she has not otherwise been granted permission to edit them.  

Since this override is intended to apply only to a staff member's own Requests, users with this permission cannot edit the Staff field of Requests that are assigned to other people: this would allow them to assign Requests to themselves in order to edit these fields.