Cut actually refers to two aspects of a diamond. The first is its shape (round, oval, princess, cushion, etc) the second is how well the cutting has been executed.
A diamond's cut will most certainly influence its fire (the lovely rainbow colours that flash from within) and brilliance (the liveliness and sparkle), as well as its perceived size and even, to some degree its apparent colour. Different cuts reflect light in different angles. A diamond must be cut in a geometrically precise manner to maximise its brilliance.
It is the technicalities of exact angles, proportions, symmetry, and polish that affect the way a diamond handles light. In short, how well a diamond is cut determines how well the diamond will shine.
In order to understand if a diamond is well cut we need to understand its proportions: whether the stone is too shallow, too deep, or just right. A well-cut stone is carefully and precisely crafted: all the facets are exactly where they should be and polished until they gleam. To be well cut, a diamond must have the right proportions, precise symmetry, and a fine polish. Diamond dealers refer to this as "make," and it is the only thing about a diamond's quality that can be controlled by man. A stone with fine make has been carefully fashioned by the cutter to maximise its potential.
Whichever your choice of cut; be it round, princess, oval or any other diamond shape, a well-cut diamond will reflect light back evenly in the face up position, with no dark areas. A well cut diamond returns the maximum amount of light to the eye as brilliance. A well cut diamond also displays dispersion or fire: spectral colours of light that add richness to its sparkle.