Difference between $\operatorname{Var}(Y)$ and $\operatorname{Var}(Y\mid X)$? What is the difference between $\mathrm{var}(Y)$ and $\mathrm{var}(Y\mid X)$? If $Y = c + \beta X$ and $\operatorname{var}(X)=\sigma^2$, won't both come out to be the same, i.e., $\beta^2\sigma^2$?

$\operatorname{Var}(Y)=\mathbb{E}(Y-\mathbb{E}(Y))^2$ is a number.  Var(Y|X) is the conditional variance of $Y$ given $X$:
$$
\operatorname{Var}(Y\mid X):=\mathbb{E}[(Y-\mathbb{E}[Y\mid X])^2\mid X]
$$
is a function of the random variable $X$.
