
Carats are the measure of weight for a diamond, and are not related to Karats which are a measure of purity for gold metal. The size of a diamond, measured in millimeters, is not the same as its carats. Think of a wide flat diamond versus a skinny long one (both would be improperly cut in this instance), the long diamond might have more weight but look smaller from the top.
Here is the standard diamond scale with all diamonds cut to the same proportions, so you can compare carats to carats:
Carat Weight Scale

The term “bigger is better” is not the only measure of a fantastic diamond. The other 4Cs (Cut, Color and Clarity) are also important factors in determining a diamond's value. You should balance the other Cs and then find the largest number of carats to fit within your budget.
Once a diamond is mounted in a setting it is more difficult to exactly grade a diamond, because of this the Gemological Institute of America will only grade unmounted diamonds. Your grading report will tell you the number of carats to the nearest 100th of a carat.
Remember: it is better to find a balance between the 4Cs when choosing a diamond, size isn't everything. The most valuable diamonds are those with the highest level of cut (refracts light brilliantly), color (meaning colorlessness), clarity (no inclusions) and carats (weight).