The on-off control method is the least expensive  closed loop system to use due to its simplicity. The on-off control  method does however tend to cause more wear on its equipment due to the  excessive amount of on-off cycles. It is also the hardest control system  to maintain a consistent process variable because the tolerances are  expanded to avoid starting and stopping of the load device. A good  example of an on-off system is a conventional oven. The heating element  is the load which has a controller that operates off of feedback from a  thermocouple. 

Proportional Control method is a way to meter the  output by monitoring the measured variable. The controller varies the  output according to the error signal produced between the setpoint and  the measured variable to mitigate error between them. For instance, the  larger the error the larger the output command. Let's say there is a  water tank that opens a valve to drain the tank at a rate of 30 GPM. The  proportional control method is used to adjust the output (pump) to  match the flow into the tank as what is flowing out of the tank. So, the  output increases to equal the 30 GPM.  

Proportional-Integral method is the same as a  proportional control with an inputted integral change to ensure that  offset is minimized. The Proportional portion controls the margin of  error but does not make up for the offset between the control variable  and the setpoint. The integral control continuously adjusts the output  reference to eliminate offset. Continuing on the water tank example in  the proportional control description, while the output has adjusted to  match the flow of water entering as what is exiting, there was an offset  created due to the slow increase of inflow. The tank is 30 gallons  under its setpoint by volume. The integral mode is used to increase the  output so that offset of the tank level is at zero (back at setpoint  pertaining to level). 

Proportional-Integral-Derivative method is a way to  produce even tighter tolerances between the control variable and  setpoint. This method is to keep the controller from overshooting by  using the derivative portion to sense the control variable's change  relative to the setpoint. This enables the controller to respond more  appropriately to ensure that the measured variable does not stray too  far from setpoint by responding to more sudden process changes. A  "boost" is given to the output to allow the proportional method to take  control quicker so that there is less of opportunity for error. This  portion can be included in the previous water tank process where the  emptying valve opens up at a drain rate of 30 GPM. The derivative can  instantly ensure that the output drives at a higher frequency so that  there is less error for the proportional method and less offset in the  integral method. 

References:

Bartelt, T. L. (2011). Industrial Automated Systems: Instrumentation and Motion Control. Cengage Limited. https://ecpi.vitalsource.com/books/9781305474277