A brilliant cut refers to a round brilliant cut diamond.  This diamond cut is considered the brightest cut.  It consists of 57 triangular and kite-shaped facets.  The top of the cut is called the crown, the middle section - the girdle, and the bottom is called the pavilion.

The round brilliant cut is the most popular diamond shape.  This demand leads to it being the most expensive diamond cut.  Another factor that influences it's price is that rough diamonds have a square profile so to cut a round stone from this shape leads to a lot of waste.