The covariance of two random variables $X_1$ and $X_2$ with mean $\mu_1$ and $\mu_2$ respectively is defined as

\begin{equation} \cov(X_1,X_2) \defined E[(X_1 - \mu_1)(X_2 - \mu_2)]. \end{equation} The covariance of a random variable $X$ with itself is simply the variance, $E[(X - \mu)^2]$

Covariance captures a measure of the correlation of two variables. Positive covariance indicates that as $X_1$ increases, so does $X_2$ Negative covariance indicates $X_1$ decreases as $X_2$ increases and vice versa. Zero covariance can indicate that $X_1$ and $X_2$ are uncorrelated.

The correlation coefficient provides a normalized view of correlation based on covariance:

\begin{equation} \corr(X,Y) \defined \frac{\cov(X,Y)}{\sqrt{\var(X)\var(Y)}}. \end{equation} $\corr(X,Y)$ ranges from -1 (for negatively correlated variables) through zero (for uncorrelated variables) to +1 (for positively correlated variables).

While if $X$ and $Y$ are independent we have $\corr(X,Y)=0$ the latter does not imply the former.