What do air traffic controllers do on a slow day?
The airspace around the world is divided into sectors within a Flight Information Region (FIR) /Air Route Traffic Control Center (ARTCC) / Area Control Center (ACC), which range from very small and tight airspace pieces to larger combinations or enroute sectors. Enroute sectors will usually be larger than Terminal Maneouvering Areas (TMA).

During peak-hours, these sectors will be handled by one controller responsible for each sector. If capacity and traffic flow permit, these sectors can be combined and two sectors will be handled by the same controller. 

Below is a picture of the Paderborn Lower Sector, ranging from GND to FL135. This is usually a stand-alone sector which only covers approaches to Paderborn (EDLP) and Dortmund Airports (EDLW).

(Image Source: Own Work - Author: SentryRaven)

If traffic permits, the adjacent sector Hamm Lower from GND to FL115, handling arrivals and departures to Münster (EDDG), will be handed over to the controller responsible for Paderborn Lower. This can also be done vice-versa, where the Hamm Lower controller becomes responsible for the Paderborn Lower sector.


(Image Source: Own Work - Author: SentryRaven)

TMA controllers not only handle IFR aircraft approaching or departing from airfields within their sector, but also any enroute traffic crossing the sector, VFR traffic requesting to cross airspaces which require prior permission and other tasks. 

As Chatty already pointed out in his answer, controllers take breaks between working their sectors and sometimes often switch sectors during a shift. A controller working Paderborn Lower during the beginning of his shift can end up working Hamm Lower towards the end of his shift with adequate breaks in between their sessions. 

See this related question:
What is the maximum number of planes a controller can control?