I'm not entirely sure about the responsibilities of mid-grade admirals. The fleet admirals usually serve on the Joint Chiefs. Perhaps there are some BBS users in the Navy who could answer this more accurately than I, but there are different commands. Surface warfare, submarine command, SEAL deployment, etc., also divided up into regions, like Pacific Command, Atlantic Command. I would hazard to guess the Rear Admirals (lower half) command individual battle groups and task forces. Rear Admirals (upper half) command the regions and zones. Vice Admirals may act as a conduit from the Fleet Admirals and Joint Chiefs to the Rear Admirals to distribute orders and information. Again, I'm pulling this pretty much out of my ass, but if I were to guess at the job descriptions for the different grades, that's how I would envision it.
Applying that to the world of Starfleet, however, would probably be broken down by neutral zone and border region defenses, as well as a separate command designated purely for exploration and another for internal research (scientific tasks, stellar cartography, archaeological and xenobiological expeditions) and infrastructure support.