A measure of the degree of association among a pair of variables.
The Pearson correlation between two random variables $X$ and $Y$ is defined as
$$ {\rm cor}(X,Y) = \frac{ E(XY) - E(X)E(Y) }{ \sqrt{ {\rm var}(X) {\rm var}(Y) } }$$
and is bounded between $-1$ (perfect negative linear relationship) and $1$ (perfect positive linear relationship). The numerator of ${\rm cor}(X,Y)$ is known as the covariance between $X$ and $Y$.
If the Pearson correlation is $0$, we say the two variables are linearly independent.
Other definitions of correlations exist and can detect non-linear relationships. For example, Spearman's Rank Correlation' and Kendall's Rank Correlation
