A random variable is a property of a random process. That is, a process in which it is difficult to predict the outcome. The process may occur many times, and each time, the random variable may have a different value.
A random variable is called discrete if it can have only countably many possible values. In other words, if the set of possible values can be listed, even if this listing continues forever.
Otherwise, the random variable is continuous. Typically, the set of possible values for a continuous random variable forms a range.
For example, the rolling of a die can be considered a random process. The number that shows on the top of the die after the roll is a discrete random variable associated to that process, and the amount of time it takes for the die to stop rolling is an associated continuous random variable.
Each random variable has an associated a probability distribution, which describes, for each value that can have, the probability that it will actually have that value (or in the case of continuous distributions, a value close to it).