Quartz Crystal 
Quartz, composed of silicon and oxygen (silicon dioxide), exhibits piezoelectric properties. 
This device generates an electrical potential when pressure is applied on the surfaces of the crystal. 
Inversely, when electrical potential is applied to the surfaces of a crystal, mechanical deformation or vibration is generated. These vibrations occur at a frequency determined by the crystal design and oscillator circuit.

Figure 1 Quartz Crystal Structure   Figure 2 Cross Section of Crystal Perpendicular to Axis ZZ 
Cutting Angle & Vibration Mode  
Cut angles differ depending upon the applications (Oscillation frequencies and electrical characteristics). The crystals used in crystal units are normally in the form of plates or elements cut from synthetic crystal. The Fig. 3 shows main cutting angles and Fig. 4 shows vibration modes, frequency range and capacity rations (typical values). Take the most popular AT-cut crystal wafer for example, it is in a plane which makes an angle of 35º15º to the Z-axis and the wager thickness is approximately 0.06mm in the case of 25MHz fundamental-wave thickness-shear vibration.