Diamonds
Cut
When we speak of cut we are more interested in the proportions of the diamond as opposed to its shape (Round Brilliant, Marquise, Pear, Princess, etc.) Every diamond regardless of its shape gets it brilliancy and scintillation by cutting and polishing the diamond facets to allow the maximum amount of light that enters through its top to be reflected and dispersed back through its top.
Color
Diamonds come naturally in every color of the rainbow. However most people are concerned with diamonds in the white range. The Gemological Institute of America (GIA) rates the body color in white diamonds from D (colorless) to Z (light yellow).

The best color for a diamond is no color at all. A totally colorless diamond allows light to pass through it easily, resulting in the light being dispersed as the color of the rainbow. Colors are graded totally colorless to light yellow. The differences from one grade to the other are very subtle and it takes a trained eye and years of experience to color grade a diamond.
Clarity
Most diamonds contain very tiny birthmarks known as "inclusions."
Carat-Weight

This is the weight of a diamond measured in carats. As the carat weight of a diamond increases so does its rarity and therefore its price.
One carat is divided into 100 "points," so that a diamond of 75 points weights .75 carats. The carat-weight of a diamond is the easiest measurement to determine. Most importantly, two diamonds can be of equal carat-weight, but their value can differ greatly due to their cut, color, and clarity.