Crystal Systems 

Cubic The cube is composed of 6 square faces at 90 degree angles to each other. Each face intersects one of the crystallographic axes and is parallel to the other two. 

Tetragonal The tetragonal system also has three axes that all meet at 90°. It differs from the isometric system in that the C axis is longer than the A and B axis which are the same length. 
Zircon  
Hexagonal In
the hexagonal system we have an additional axes,
which gives the crystals six sides. Three of
these are equal in length and meet at 60°
to each other. The C or vertical axis is at
90° to the shorter axes. 

Trigonal Again, the trigonal system is a subsystem of the hexagonal. Most gem references will list these as hexagonal. 
Amethyst 

Orthorhombic In this system there are three axes, all of which meet at 90° to each other. However, all the axes are a different length. 

Monoclinic . The above crystal systems all have axes sides that meet at 90°. In the monoclinic system all the axes are different lengths. Two of them, the A and C axes, meet at 90°, but the third one does not. 

Triclinic In this system all the axes are different lengths and none of them meet at 90°. 
Turquoise 