The cut of the diamond determines how brilliant and fiery it is because it determines how the light falls into the gemstone and is reflected back; therefore, the cut should not be too shallow or too deep. Some jewelers differentiate the cut of the diamond from its shape.
The clarity of the diamond refers to how many flaws it has. An observer shouldn't be able to see any flaws in a diamond with the naked eye.
A diamond that's said to be colorless probably has a slight yellow tint. This too should be hard to see with the naked eye. Colored diamonds, called fancies, are also available for purchase. They range from pink to blue to brown to black.
Carat size refers to the size of the diamond. Large diamonds usually cost more than small ones though a large diamond with many flaws and a bad cut can cost less than a diamond that's much smaller and flawless.