The acronym “PID” stands for “Proportional, Integral, Derivative.” PID itself is actually a set of equations used in many industrial control applications - from aerospace engineering to energy generation to beer brewing - the small electronics box that is referred to as a PID controller is simply a physical representation of the algorithm. This algorithm allows for precise control of multiple system variables, like the temperature of an espresso brew boiler.
Before PID controllers, espresso boiler temperature was controlled by thermostats or pressurestats - small, simple mechanical devices that turn the boiler on when it dips below a certain temperature, and back off once it reaches the desired temperature. These systems worked, but they weren’t very accurate. The boiler had to cool down below a set temperature before turning on and continued heating up slightly past the shutoff point at the top end, making temperature stability quite volatile and difficult to precisely control.