"Carat Weight" is simply the weight of the diamond. The following table shows, roughly, the appearance of different carat weights:
By definition, 1 carat is exactly 200 milligrams. Since most diamonds sold in the jewelry industry weigh less than 1 carat, the carat is usually subdivided into "points." There are 100 points in 1 carat, so that a diamond weighing 3/4 carat would be a "75 point diamond. In summary:
1 carat = 200 milligrams = 100 points
Don't confuse the weight of a stone and its size. This argument is similar to that of cereal boxes or potato chips, where "contents are sold by weight, not volume." Also, don't confuse the carats of a diamond with the carats of a different stone. Because different minerals have different densities, a 1-carat diamond will have a different volume (a.k.a. size) than, say, a 1-carat ruby. Another common source of confusion: The karats used to measure the purity of gold have nothing to do with the carats used to measure the weight of a diamond. Notice the difference in spelling.
If you're shopping for a diamond and you're unsure of a given stone's carat weight, simply ask the jeweler to weigh the stone in your presence. After making sure that the balance reports "0" when there is nothing on it, place the unmounted stone on the balance, take the number of milligrams reported by the instrument, and divide by 200 to get the carat weight of the stone.
Shopping for a diamond strictly based on size would be similar to purchasing a car solely on its size without considering features and styling. If size (carat weight) is your most important attribute, that is perfectly acceptable -- but do consider the other 3 Cs to make sure you understand your tradeoffs.