What is the expectation of the variance?

The expectation of the variance is simply the variance itself: $\mathbb{E}[\mathrm{Var}(X)] = \mathrm{Var}(X)$.

Variance is defined as $\mathrm{Var}(X) = \mathbb{E}\bigl[(X - \mathbb{E}[X])^2\bigr]$ and measures how much a random variable deviates from its mean.