In terms of espresso technology, a simplified definition of PID = an electronic brew temperature controller. There's tons of information on
proportional
integral
derivative controls (PID). Below are a few introductions:
The subject of PID'ing Silvia is well worn on CG, alt.coffee, and of course
"PID" is mentioned frequently on HB. Below is a small sample:
By the way, I got better quality search results on google from "proportional derivative controller" rather than "PID."