Effect of P,I and D terms on output response in PID controller?
P, or proportional is a fixed gain response, if the input moves below set point, the output moves up by gain * input error, "P" is that gain

I, or integral is literally an integrator so used to react to low frequency offsets, a tiny error positive or negative keep on summing up skewing the output to reduce the offset that was outside of what the proportional response gave, "I" is a scaling factor before the error is summed to the output

D, or the derivative, is a differentiator, it reacts to high frequency offsets, it essentially outputs the rate of change of the input, "D" is the scaling factor before this slope rate is summed to the output, 

Integral can be viewed as correcting for past errors in a sense, but its not quite right, more like its an offset trim for the proportional output center point due to the limitations of such a simple control loop, 

Derivitive can be veiwed as correcting for future errors, again not quite right, but close enough, if the input starts dropping fast it cranks open the output much faster than the integral or proportional would, equally tempers how quickly the input can change over a given time period, 

Integral is the low pass filter, Derivitive is the high pass filter, in an op amp circuit you might think of it like a compensation network to improve the stability of a circuit,