In this easy to read and much popular article, Tim Wescott bridges the gap between theory of PID controllers as taught in coursework, and how they are used in the industry in hardware as well as software. He vividly explains how to think about Proportional, Integral and Derivative control in simple terms, by linking the physical significance or interpretation to their mathematical concept. For example:

We know that proportional control deals with the present behavior of the plant, and that integral control deals with the past behavior of the plant. If we had some element that predicts the plant behavior then this might be used to stabilize the plant. A differentiator will do the trick.

A derivative in calculus (differentiation) means rate of change at a given time. This means, that at any given time, we can know whether we are converging or diverging from the desired value. Accordingly, we can feed back the signal in an additive manner (positive feedback) or in a way that reduces the input (negative feedback)