A piezo disc with only 2 terminals in or out of an enclosure is a transducer that plays an external AC signal.
A piezo with 3 terminals and a transistor oscillator beeps when it is powered from DC.

A crystal oscillator uses the crystal as an inductor and two capacitors produce tuning and feedback.

 

A piezo disc with only 2 terminals in or out of an enclosure is a transducer that plays an external AC signal.
A piezo with 3 terminals and a transistor oscillator beeps when it is powered from DC.

A crystal oscillator uses the crystal as an inductor and two capacitors produce tuning and feedback.

You are describing a Colpitts circuit, one of several kinds commonly used.

 

Beepers are the active ones which include an oscillator, there are passive piezo speakers which have a narrow resonant frequency that can be used to outputting PWM.