The diamond cut is responsible for a diamonds brilliance or how much it sparkles and refracts light.
The diamond cut is closely related to the shape of the diamond. Some overall shapes lend themselves to better refractive properties.
There is no doubt that a well cut Round Brilliant Cut (RBC) diamond is the optimum shape for overall brilliance. However with the advent of computer design, modelling and indeed even cutting, it is possible to have excellent brilliance and accomodate more unusual diamond shapes.

A diamond cut with a pavillion depth that is too deep or too shallow will refract light out of the sides instead of the table (top).
Symmetry and the angle of the cuts are all important. An off center cullet (viewed from the top, sloping table, or chips will all have an effect on the refractive qualities of your diamond.
Computer technology has allowed for very precise cutting and designs that optimise refraction.