On-Off control is the simplest form of feedback control. An on-off controller simply drives the manipulated variable from fully closed to fully open depending on the position of the controlled variable relative to the setpoint. A common example of on-off control is the temperature control in a domestic heating system. When the temperature is below the thermostat setpoint the heating system is switched on and when the temperature is above the setpoint the heating switches off.
There is, however, a bit of subtlety applied in practical on-off systems. If the heating switches on and off the instant the measured temperature crossed the setpoint then the system would chatter - repeatedly switch on and off at very high frequency. If this happened the boiler wouldn't last very long! To avoid chattering, practical on-off controllers usually have a deadband around the setpoint. When the measured value lies within this dead-band the controller does nothing - its only when the value moves outside that action is taken. The effect of this is to introduce continuous oscillation in the value of the controlled variable - the large the dead-band the higher the amplitude and lower the frequency.
A simulation of an on-off controlled system can be found here.
Although on-off is a very cheap form of control it is rarely used in process control applications because of the oscillation it causes in the controlled and manipulated variables. In a connected process these oscillations would be propagated right through the system.