Unit cells and crystal systems

A unit cell is a region which can be repeated to build up an entire crystal. The smallest unit cells possible are preferred. For example, b is preferred to a in the diagram below:

However, sometimes larger unit cells are used in order to make symmetry features more apparent.

All ordered structures adopted by compounds belong to one of seven crystal systems:

The letters in the diagram are called unit cell parameters, composed of a b c and angles α β γ. 

The angle between a and b is γ.

The angle between b and c is α.

The angle between a and c is β.

As shown in the pictures, not every unit cell uses all six parameters. 

Rather than memorising the shapes it easier to remember the three vectors used to produce these shapes, which correspond to the polyhedra above:

The lines are in 3d, as in an octahedral molecule from A-level:

Cubic: All vectors perpendicular to eachother. All vectors are equal length.

Tetragonal: All vectors perpendicular to eachother. Only two vectors are equal lengths.

Orthorhombic: All vectors perpendicular to eachother. All vectors are unequal lengths.

Monoclinic: One vector perpendicular to the other two, other two are at an oblique (not a multiple of 90°) angle to eachother. All vectors are unequal lengths. 

Triclinic: All vectors intercept at different oblique angles. All vectors are unequal lengths.

Rhombohedral: All vectors intercept at the same oblique angle, all vectors at the same length.

Hexagonal: This one requires four vectors. Three are the same length in the same plane at 60° to eachother, and one vector is perpendicular to the three and the same or a different length:

I suspect the best way to learn these is to draw the vectors and apply them to the solid shapes.