What are the essential elements of a control tower?
The theoretical bare minimum would be a light gun to give light signals to aircraft. A more generally acceptable minimum is a two-way radio to do radio communications with aircraft. 

From there up other systems can be added. The list below gives some examples but is not exhaustive. 


A telephone. Always useful when you want to get in contact with other parties on the ground (e.g. air traffic control centre) or other parties with you. 
Meteorological instruments can provide current wind and visibility at the runway to the tower controller. This is not directly to support the Air Traffic Control function, but to provide information services to pilots. 
Airport lighting control: runway (and taxiway) lights are usually controllable from the tower. These systems can become quite complex on larger airports
A display with a feed from an approach radar. This present the air traffic situation around the airport to the controller. Whilst on larger airports the approach controllers are usually not located in the tower itself, smaller airports often combine tower &amp; approach control. Even when approach controllers are physically located elsewhere, the approach radar assists tower controllers as well in building a clear mental picture of the situation. 
An emergency button that makes the fire department jump to life.
Surface radar display. This presents the ground traffic situation to the controller. Often not found on small airports, but essential for efficient operations at busy airports, especially in low visibility.  
More radio's. To split tower control who do the runways and ground control who do the taxiing aircraft. Further divisions can be made. 
A system that shows which runways are active, which are closed for use by the airport authority (e.g. for maintenance) 
Electronic flight strip system. The traditional paper strips are now often replaced by a digital equivalent. These E-strip system usually communicate with the whole flight data processing system, correlating strips to radar tracks. 
Gate allocation system. It's always nice to now where those arriving aircraft should be directed once they are of the runway. 
Gate video systems showing the situation at the gate. (to bust pilots that say they are ready for engine start / pushback whilst they are still loading cargo :-) )
Departure management system. When you have a complex airport with multiple runways and a maze of taxiways, a DMAN may help to get aircraft airborne at the right time in the right sequence in the right direction.
Arrival management system. With many aircraft arriving from all directions, headed for different terminal / gates it helps to have an AMAN to avoid overloading your Terminal Manoeuvring Area with incoming flights competing to be first.  
Integration of the Surface radar, DMAN and Gate Allocation System with the lights control system. The level of integration varies. In the most extreme case pilots just have to follow the taxiways that light up green and stop at red stop bars. This avoids a lot of radio communications and confused pilots who didn't exactly remember a taxi clearing involving a dozen of taxiway letters.