The cut refers to the positioning of each of the facets on a diamond.
This is the only one of the Four C's that is determined by anything other than nature. A master diamond cutter will look at a piece of diamond rough over and over until he finds the best possible way to cut the stone. The correct positioning of each facet reflects light from one to another, causing white light to reflect (generating brilliance) and disperse into it's spectral colours (generating fire).
These flashes of brilliance and fire give the stone it's beauty. The other C's count for very little if they cannot be unlocked by the correct cut.
Sometimes stones are cut to save carat weight but in so doing upset the balance of a stone, detracting from it's sparkle. Some stones are cut too shallow. These diamonds let light leak out through the side of the stone. A stone like this will look bigger than a correctly cut stone of the same weight. This may seem an advantage but because of the light lost through the side the stone will not sparkle remotely like a well cut stone.
Stones that are cut too deep are cut this way in order to retain unnecessary carat weight. The stone may weigh 1.50ct but looking at it in a setting it may only look like a 1.00ct stone. A stone like this will also have a dark centre as light will leak out through the bottom of the stone.