From this post, given a collection of sets $\mathcal{C}_0$, we can construct $\sigma(\mathcal{C}_0)$ through an iterative process, where for each step we let $\mathcal{C}_\alpha$ be the collection of all countable unions and complements of sets from $\bigcup_{\beta < \alpha}\mathcal{C}_\beta$.
Let $\mathcal{C}_0$ be the collection of all intervals on the unit interval. What would be an example of a set that is in $\mathcal{C}_2$, but not in $\mathcal{C}_1$?
The Cantor set has these properties. It is the complement of a countable union of intervals so it is the complement of a set in $\mathcal C_1$. But it is not in $\mathcal C_1$ since it is neither the complemenet of an interval nor a counteble union of intervals. [Recall that $C$ has no interior and it is uncountable].