The prefix "Piezo" is derived from the Greek piezein, which means to squeeze or press.
In the case of a crystal, it is the electrostatic effect on the material. Crystals are mostly made of quartz, which has a crystal lattice that is very stiff and returns most of energy back from a flexing action. Most all materials have some piezo-electric effect but you need a material that has the stiffness and quality factor to 'ring' for a period of time. A string of hair being pulled by a static charged comb is demonstrating piezo-electric effect but a hair does not have sufficient stiffness to spring back and forth for very much time.
Now think of a wine glass being 'twanged' by a flick of a fork hitting the edge of the glass. The glass will vibrate for some period of time, due to the mechanical impulse hit, at a particular audio tone that is the natural 'ringing' frequency (resonance) of the glass. This is very similar to a quartz crystal.
You may have seen demonstations of a wine glass breaking due to an audio tone from an audio loudspeaker (or very loud singer). The audio tone is exciting the wine glass at its natural resonant frequency. If the excitation is strong enough it will excite the glass (flex the glass) beyond its breaking point. The same thing can happen to a quartz crystal in an oscillator circuit.
As to A.C. or D.C., a quartz crystal must have some external excitation to get the oscillation started. This initial excitation can be from the transient ('twang') of initially applying the power supply (D.C) to the oscillator circuit, or low level electrical noise within the amplifier (transistor, etc.). The crystal will begin to ring at its natural resonant frequency which is amplified and reapplied to further increase the crystal's ringing. Eventually the circuit will reach a steady state of constant amplitude, hopefully below the point where crystal fractures.