What types of air traffic controllers are there?
In Canada (and I believe similarly in the United States), there are four ratings which can be applied to an Air Traffic Controller license:


Airport Control
Terminal Control
Area Control
Oceanic Control


Within each of those ratings, the responsibilities are sub-divided a bit based on operational needs. For example, controllers at a tower (with an 'Airport Control' rating) would tend to be qualified in all aspects of tower operations including:


Clearance Delivery (issuing IFR clearances or assigning transponder codes)
Ground Control (issuing clearances for aircraft to move to/from the apron and runway areas via a taxiway system
Air Control (issueing clearances to aircraft in the runway environment and in the vicinity of the airport)


Terminal Controllers (if a terminal control area exists around the airport in question) would be responsible for flights in the immediate vicinity of the airport to approximately 20-50 nautical miles. Their responsibilities may include multiple small airports, just a single large airport, or even just one specific operation at an airport (such as arrival control or departure control). These controllers are usually allowed to apply a smaller amount of lateral separation between aircraft due to the dense nature of the terminal environment and the specific nature of their role.

Once the aircraft leave the terminal (if one exists), the Area Controller-sometimes called enroute control-takes over. They are responsible for anywhere between several hundred square miles of airspace to over several thousand square miles in the case of sectors in northern Canada. 

In the case of an aircraft transiting the ocean, a specific type of area controller called an Oceanic Controller takes over. They possess slightly different training in order to safely separate aircraft over the ocean in non-radar airspace.