Crystal systems describe the shape of crystal structures. By reducing the system to the smallest amount that can still be considered a crystal of that type chemists have found there are only 7 fundamental shapes (called crystal systems). The length of the sides, and the angles that join them describes each system. Consider a simple square box. Each of the axes length, width, and height, are equal and they make right angles with each other, this is Cubic crystal system. If it is stretched along one direction (lengthen the box) it is Tetragonal, stretch it in two dimensions and it is Orthorhombic. If now this badly stretched box is skewed (shear the top and bottom) so that the tops no longer make right angles it becomes Monoclinic, if it is skewed again in another direction so no face is at right angle to any other it becomes Triclinic. Ok now back to the original cube (all sides equal, all angles equal to 90 degrees) if this cube is skewed exactly the same amount in both directions it is Rhombohedral. Lastly if the original cube is stretched and then skewed to exactly 120 degrees then it is called Hexagonal.

Here in a nice tabular form are the different crystal systems.
1)From c programming != means not equal.
2)d = angle between b and c, e between c and a, and f between a and b.

Name:         Axes:      Angles(in degrees):
Cubic          a=b=c      d=e=f=90
Tetragonal     a=b!=c     d=e=f=90
Orthorhombic   a!=b!=c    d=e=f=90
Monoclinic     a!=b!=c    d=e=90 f!=90
Triclinic      a!=b!=c    d!=e!=f!=90
Hexagonal      a=b!=c     d=e=90 f=120
Rhombohedral   a=b=c      d=e=f!=90