PID is a controller that uses
Proportional,
Integral, and
Derivative
equations to controla linear system. Cruise control in a car is a good example of how PID is used; the car knows how fast it is going, how fast the driver wants to go, and the computer does calculations to determine how much power to send to the wheels.
Wikipedia has some good information about how PID works, some information on manual tuning, and how adjusting each parameter changes how the system reacts.
Bang-bang control a quick implementation PID, with the P gain set high and ID set to zero, and with only on and off output instead of the full range of values the motor controllers can accept.
Theincreased complexity of PID comes with the increased control over the system.
We used both bang-bang and PID a few days ago on our shooter motor; both worked however PID was smoother and allows you to set any speed instead of having to hard code values.