What is the difference between the 3 circuits? (OP-AMP)
Assuming the non-inverting input (+) to the op-amps are wired to ground then these statements should be true:

The top right is an inverting amplifier

The top left is an inverting amplified integrator

The bottom is an inverting integrator

When should you use them? Well.. either just like that, integrate / amplify a signal from a sensor, or use it as a part of a filter to remove some frequency components you wish to remove. Turn a sine wave into a cosine wave. A part of a PID regulator (especially the P and I part). Or tons of other uses. 

But when you earn more experience you will realize that you will make these as you need them. It's as if someone is asking you about multiplication and division and when to use them. Well.. you use them... when you need them to solve your particular task. It will be obvious what to use and when to use it. Just as if I would ask you what the area of a 40 meter by 30 meter rectangle is. You would use 40×30.