How Do Integration and Differentiation Function in Electronic PID Controllers?

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I'm curious about what does really integration and differentiation mean in electronic stuff like PID controller.

In the frequency stabilization of diode laser, the cavity length is changing by applying a sinusoidal voltage to the piezo attached to it. By turning the piezo integrator of the PID controller, I'm able to lock the frequency to a fixed frequency.

The example may not necessary point to PI controller.

Differentiation means differentiation of a curve? This also mean the same thing in PID controller? Let's say a laser give an absorption wave profile of an atom, differentation means to differentiate the graph shown on the oscilloscope?

Thank you.
 

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For PID control this differential part is for when you go outside your proportional limits. The error signal will have a gain associated with it causing the controller to accelerate the error signal (difference in actual from set-point) to zero. Note; the integral limits will be inside the proportional limits.