The cut is not the same as the shape. Cut classifies the diamond as how well the light enters and reflects inside the diamond, which gives a more beautiful diamond the better the cut.
The color of the diamond is determined by the environment the diamond is created in. The most popular and beautiful color is the colorless or white color, but some people like a more yellowish color. Bright yellow diamonds are classified under fancy diamonds, whereas faint colored stone are labeled alphabetically. The brightest white color is called D and the more yellow it becomes the further down the alphabet it goes.
Clarity is determined by how many impurities the diamond contains and how visible they are to the naked eye. Impurities are natural and some people may like the authenticity, but in general the better the quality the more expensive the stone.
Carat is a weight unit and is the way you measure the size of diamonds. The bigger the stone, the higher carat.