What is a transitive class? What is the significance of this notion for models of set theory?

Prove that for any set $x$ there is a least transitive set $\mathrm{TC}(x)$, the transitive closure of $x$, with $x \subseteq \mathrm{TC}(x)$. Show that for any set $x$, one has $\mathrm{TC}(x)=x \cup \mathrm{TC}(\bigcup x)$, and deduce that $\mathrm{TC}(\{x\})=\{x\} \cup \operatorname{TC}(x)$.

A set $x$ is hereditarily countable when every member of $\mathrm{TC}(\{x\})$ is countable. Let $(\mathrm{HC}, \in)$ be the collection of hereditarily countable sets with the usual membership relation. Is HC transitive? Show that $(\mathrm{HC}, \in)$ satisfies the axiom of unions. Show that $(\mathrm{HC}, \in)$ satisfies the axiom of separation. What other axioms of ZF set theory are satisfied in $(\mathrm{HC}, \in) ?$