coming from the double bass side, mostly, I have some experience with piezos. Also I used to have a e-bass with a piezo pickup.
They work by sensing changes vibration--a physical change in the crystal material they are made of. They often have to be under some pressure to work well, like under the saddles of a bridge.
Unlike a magnetic pickup, they have an extremely wide frequency response and so they get much more of what the string is doing. More lows, more highs. That's the main reason why they are used much more in acoustic instruments.
Downside is they havehave, to my ears, a nasty "quack," a hard to describe and hard to get rid of timbre that shouts "piezo" right away. You can minimize this--see below
They also have a high impedance. The long and the short of it is if you plug them directly into your good old bass amp, they will sound ear bleedingly awful. You need a preamp in front of them, almost always, and a preamp with a high impedance input. For double bass, I have a preamp which has switchable input impedance. A lot of double bass player use this nifty device:
https://sites.google.com/site/hpftechllc/home/hpf-pre
It's designed to work well with high impedance piezos.
Getting the impedance matched right is in my experience the single most important thing with piezo pickups. If it matches well, you still can tell it's a piezo pickup, but it's not so offensive.