Both questions are surprisingly deep. Lets take the easier one -- on the quartz crystal -- first. Anytime you put an electric field across a material, you create a force on the electrons and protons in the material-- in opposing directions. This can have many effects,depending on the material, but for most materials there is a subtle shift in the average positions of some electrons relative to their"rest" positions. This motion sets up an opposing electric field (called the material polarization). Usually, this field simply reduces the total apparent field in the material -- the ratio of this new field to the external charge is called the dielectric constant for the material.In the case of Quartz and some other materials, the electron drift causes the material to adjust slightly the spacing of its crystal lattice -- i.e. it changes shape, proportional to the applied field.Usually this is a very small shift -- but it is a reversible process --if you squeeze the material -- you also get an apparent electric field on the outside. The fancy name for the phenomena is piezoelectric effect and it is related to symmetry properties of the underlying crystal or material.

Quartz is used in frequency stabilization because it can be fabricated into devices which 'ring' at a particular frequency.

Some materials are piezoelectric. That means that bending or stretching or compressing the material creates an electrical current. And the vice versa is true--an electrical current causes the material to bend, stretch, or compress. Quartz crystals are a piezoelectric material.

So, what people do to make them vibrate is apply a voltage, which distorts the crystal (kind of like applying a force to compress a spring), and then when the voltage is removed, the crystal springs back into place, which creates an electric field. The combination of these effects creates a kind of circuit that vibrates with a specific frequency (called the resonate frequency). The circuit is called an RLC circuit, and I know how to use them, and I the math to describe them, and I know that they resonate.