The Law of Total Variance decomposes the variance of a random variable Y into two parts:
$$ \mathbb{V}\text{ar}(Y) = \mathbb{E}( \mathbb{V}\text{ar}(Y \mid X) ) + \mathbb{V}\text{ar} ( \mathbb{E}(Y \mid X)) $$
Intuitively, the total variance of $Y$ is equal to the sum of average variation in $Y$ within groups that are defined by $X$, and the variation in average $Y$ between these groups.
This page presents a short proof, based on the Law of Iterated Expectation. It works by replacing two terms with an equivalent expression using this law, then adding an subtracting the same term and regrouping.