Carat Weight is the weight of the rock. Carat (ct) is a unit of measurement for weight, and one carat is 0.2 grams. Another unit used in the diamond industry is "points", and 100 points equals 1 carat.
A common misconception is that weight equals size. Let's say a poorly cut diamond ("A" in the diagram below) weighs 1 carat, but it carries dead weight down at the pavilion (the bottom part). Its poor proportions can actually make it less sparkly. If you shave off some dead weight from its pavilion (becomes "B" in the diagram below), it becomes lighter in weight, but its table (the top surface) still is the same size and same diameter. So looking from the top, when it is placed inside a ring setting, it will look sparklier and still look the same size.

So why would the diamond cutters make a diamond that looks like "A"? Well, Carat Weight has been taken by the general public as a number that is the bigger the better. The jewelers can probably charge more with "A" than with "B". The thing to keep in mind here is that Carat Weight of a diamond should be regarded alongside with information about its cut.
Many of the fancy-shaped diamonds are elongated in shape, and they could appear larger than a round diamond comparable in carat weight.
Magic Sizes
Magic Sizes Diamonds are those that weigh-in exactly at or near a major size category, eg. .25ct (1/4 ct), .33ct (1/3 ct), .50ct (1/2 ct), .75ct (3/4 ct), 1 ct. etc. Many shoppers want those sizes because they are nice, even numbers that everyone can understand very easily, a lot more than, say, .68 ct. As a result, shoppers tend to pay a premium for the magic sizes diamonds.
A .68 carat diamond could look as big as a .75 carat to the eye, especially if you are not putting the two side-by-side to compare. All else (cut, color, clarity) being equal, the .68 ct could save you a bunch.
Total Weight
When you see a diamond ring listed as "2 CT. T.W.", and it's a lot cheaper than the 2 carat diamond that you've been shopping around for, don't be overjoyed yet. T.W. stands for "Total Weight", and "2 CT. T.W." means there are multiple diamonds on the ring, and all the diamonds on the ring weigh 2 carats in total. It could very well be a 0.5 carat stone in the center, and fifteen tiny 0.1 carat stones all around the ring. Or it could be a 1 carat stone, with two 0.5 carat side stones.
Of course, a 2 carat stone is more expensive than 1 carat stone and two 0.5 carat side stones together. Bigger stones are a lot rarer, and it's harder for the cutter to cut away the impurities and inclusions of a rough diamond while maintaining its weight.
Honesty is a virtue in a successful marriage, so when you tell your future bride that the ring is 2 carat, you'd better clarify whether it is 2 carat T.W. or a 2 carat center stone :-).