The 4 Cs of a diamond are Color, Clarity, Cut, and Carat.
Color is the first thing that you should take into consideration. Diamonds come in multiple colors, the clearer and more colorless a diamond the higher the receiving grade on the GIA scale. What you want is a near colorless diamond. The less color the better as can be seen in the scale below:
The second factor you should look at is the Diamonds 'cut'. Basically the better the cut the better the diamond. A poorly cut diamond will lack radiance even if the clarity and color are perfect.
A colorless diamond with a clear cut is very important for the 'clarity' of the diamond. Each diamond has flaws, flaws can include minerals left behind in the diamond, obviously the less flaws, scratches, or minerals left the higher the clarity of the diamond.
The last factor to be looked at is the 'carat' weight. You can buy a huge carat but if the color, cut, and clarity are not taken into consideration, then the diamond will be considered flawed. Carat should be taken into consideration with the other factors to make the best overall purchase. Carat also does not mean the size is bigger, simply that the diamond weights more.