A discrete variable consists of separate, indivisible categories. No values can exist between two neighboring categories. For a continuous variable, there are an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts.
Gravetter, F. J. & Wallnau, L. B. (2013). Statistics for the behavioral sciences (9th ed.). Belmont, CA: Wadsworth.