In statistics, a random variable is a characteristic, measurement, or count that changes randomly according to a certain set or pattern. Random variables are usually denoted with capital letters such as X, Y, Z, and so on.
In math you have variables like X and Y that take on certain values depending on the problem (for example, the width of a rectangle), but in statistics the variables change in a random way.
By random, statisticians mean that you don't know exactly what the next outcome will be but you do know that certain outcomes happen more frequently than others; everything's not 50-50. For example, when a basketball player tries to shoot baskets, it may not be a 50% chance he (or she) will make the basket and a 50% chance that he (or she) will miss. For some players, it may be more like a 5% chance of making it and a 95% chance of missing it.) You can use that information to better study data and populations and make good decisions. (For example, don't put those players in your basketball game to shoot free throws.)