A random variable $X$ is called continuous if its set of possible values is uncountable, and the chance that it takes any particular value is zero ($\text{P}(X = x) = 0$ for every real number $x$). A random variable is continuous if and only if its cumulative probability distribution function is a continuous function.
From Mood et al. (page 60, 1974):
"A random variable $X$ is called continuous if there exists a function $f_{X}(.)$ such that $F_{X}(.)=\int_{-\infty}^{x}f_{X}(u)du$ for every real number $x$. The cumulative distribution function $F_{X}(.)$ of a continuous random variable $X$ is called absolutely continuous".
Excerpt reference: Glossary of Statistical Terms from berkeley.edu
However, the term is also commonly used for variables that can take on a great many values, such as IQ.
