Short for piezoelectric crystal. It's a material that produces a current when compressed, so a bass fitted with a piezo bridge pickup can directly translate the strings vibrations into a signal.
The sound a piezo pickup gives is very acoustic-sounding (if a little nasal)
Standard magnetic pickups translate the strings vibration into a signal by induction of a current in a coil of wire around a magnet.
When it comes down to it, you don't really have to understand the theory... just think of it as two options that give different sounds.
On a musicman bass, the piezo pickup is fitted in addition to the standard pickups with a blend control, that can mix the two signals, so you can choose to have an all-standard sound (as if the piezo wasn't there at all), an all piezo acoustic sound, or a mix of the two.