I'm rusty with this stuff but I'm pretty sure your guess is correct, a countable model is one with countably many objects.
For other readers who might not be familiar, I'll mention that Skolem's paradox is about how there are countable models of set theory, and yet it is a theorem of set theory that uncountable sets exist, so these countable models must contain sets that are uncountable according to the model.
I think it seems less paradoxical if you think of it like this: in order for a set to be countable, there needs to exist an injection from that set to the natural numbers. So a countable model can have a set that internally looks uncountable: there is in fact an injection from that set to the natural numbers, it's just that the injection isn't included in the model.
For other readers who might not be familiar, I'll mention that Skolem's paradox is about how there are countable models of set theory, and yet it is a theorem of set theory that uncountable sets exist, so these countable models must contain sets that are uncountable according to the model.
I think it seems less paradoxical if you think of it like this: in order for a set to be countable, there needs to exist an injection from that set to the natural numbers. So a countable model can have a set that internally looks uncountable: there is in fact an injection from that set to the natural numbers, it's just that the injection isn't included in the model.