Carat is specifically a measure of a diamond's weight (when the diamond is placed on the scale), but it may not always accurately reflect a diamond's size. Some diamonds can be too "deep" and look smaller, and some may be cut better and look larger than they weight. A good analogy might be a human's height and weight: 2 people can have the same weight in lb, but one can be a lot taller than the other, so they will appear very different.
People tend to evaluate diamond size by viewing it from the top because that is how diamonds are presented to us when set into a ring.
To understand diamond size, carat weight should be considered in conjunction with the distance in millimeters across the top of the diamond. Some shapes look larger than others having the same carat weight: Pears, Marquise and Oval are the "longest".
One Carat is 100 points, so 50 points is 1/2 of a carat. The 1.56 carat is 156 points and so on..