Piezo refers to the material of which the speaker is made. Piezo is a ceramic crystal that is used in many different applications ranging from electric lighters and fire alarms to echo scanners. When compressing a piezo crystal, a voltage appears on its terminals. When applying a voltage, the crystal deforms which in case of a piezo speaker, produces a sound that we can hear. You can also use a piezo crystal as a sensor. That’s why piezo crystal or piezo transducer is actually a better name.
The piezo speaker has a high impedance (resistance for AC signals) and can be driven by an Arduino or small transistor easily. The frequency response is usually limited to a few kHz. So don’t use a piezo speaker for frequencies lower than 500Hz or so. The advantage of a piezo speaker is that these speakers can be very thin, are magnet free, and moisture proof. Most common use for piezo speakers is as buzzer.
If you use a piezo transducer as sensor it translates mechanical deformations into measurable voltages. Be aware: these voltages may be high enough to destroy the input of a micro controller it is attached to. Most common usage is as vibration or tap sensor.