A full clear of an instance occurs when all trash and bosses, including those which are optional, have been killed. A full clear is often desired by persons who are grinding faction reputation gains, such as through Championing. It should be noted that in an end-game PvE raid setting, the term full clear might be applied if all raid bosses have been killed, even if some optional trash remains.
By contrast, a partial clear implies some trash or bosses have not been killed. In the context of a dungeon setting, this usually means optional mobs have been skipped. However, in the context of end-game raiding this often means the raid group has not yet progressed to the final boss, or has skipped optional raid boss encounters.