Back to basics....
There are three main types of air traffic control - aerodrome, approach and area - though in other parts of the world they can have different names.
Aerodrome (callsign Tower and, if there is a separate position for ground movments, Ground) controllers control the taxiways and runways at an aerodrome, and the airspace "in and in the vicinity of the aerodrome traffic zone". In reality, that's airspace below 3,000 ft and out to 10 miles or so from the airfield. This is only at airfields with the traffic to warrant having a control tower. Smaller airfields may have an advisory and information service with no control authority, or they may have no one at all.
Approach (callsign Approach or, if using radar, Radar) controllers control inbound traffic out to 40 miles from the airfield and sequence them on to final approach, at which point they will be handed over to Tower. They will also control overflights operating at levels which will conflict with their traffic and departing traffic if it will conflict with arrivals. They can also give a service to aircraft outside controlled airspace within 40 miles of their airfield.
Area control (callsign Centre) control departing traffic from a few miles out from the runway IF it does not conflict with arriving traffic being controlled by Approach/Radar, and they control en-route aircraft.
In central Scotland, there are towers with on-site approach radar at Edinburgh, Glasgow and Prestwick (and RAF Leuchars who give a service entirely outside controlled airspace). Dundee tower does non-radar approach control as well as tower control.