I looked through all the other questions about countable models but none of them seem to really answer my question. I understand countability as a property from within a model - i.e. A set is countable if there is a bijection in that model from the set to the naturals. However, how do we define countability of a model?
A model of anything is a set, together with some extra stuff, with some properties - e.g. a model of ZFC is a set $M$ together with a binary relation $E\subseteq M^2$ satisfying the ZFC axioms.
A model is countable if, well, it's countable: if its underlying set is a countable set. Note that internally, of course, a model of ZFC won't think that it is countable, but I'm talking about actual countability, and there are (under reasonable assumptions) countable models of ZFC.