Variables, in the simplest of terms, are a measurement defined by a number. Variables can be either discrete or continuous, depending on the nature of what is being measured. It is important to know whether a variable or a set of variables is discrete or continuous because it will determine how the statistics derived from each can be analyzed.
A discrete variable can only be a number in a finite and countable set of values. For instance, if you are measuring the number of times you get heads when flipping a coin 10 times, you are dealing with a discrete variable, since it is finite and countable.
While a discrete variable is finite and countable, a continuous variable is one with a range of possibilities. Examples of a continuous variable are time and weight. Unlike a discrete variable, a continuous variable can be infinite and uncountable.
Discrete variables can be used to calculate the average of countable items. For instance, if you wanted to know the average number of touchdowns an NFL team scores each season, add up the total number of touchdowns for each team, which is a discrete variable, and divide by the number of teams.
Another use of discrete variable is in surveys in which the surveyor wants to quantify a quality, such as an the opinion of a product. The survey takers would be asked to rate a product with the numbers 1, 2, 3, 4 and 5, with 1 being the lowest quality rating and 5 the highest. The numbers in the ratings are discrete variables because they are finite and countable.