What is a class "in" a countable transitive model of ZFC?

Question

I'm reading Jech's "Set Theory" and on p235 he describes forcing with a class of conditions. He starts talking about the "classes in" a countable transitive model $M$, and I'm wondering what he means by this. My guess is that it's something like a set $X\subseteq M$, such that $V^M_\alpha \cap X \in M$ for every $\alpha$. Or perhaps it's more demanding: a subset of $M$ definable from parameters in $M$? Assuming it's one of these, is there something wrong with the other notion class for the purposes of class forcing?