In mathematical logic, suppose T is a theory in the language $L = \langle \in \rangle$ of set theory.
If $M$ is a model of $L$ describing a set theory and N is a class of M such that $ \langle N, \in_M, \ldots \rangle$ is a model of T containing all ordinals of M then we say that N is an inner model of T (in M). Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension of V. (inner model, Wikipedia)
Question is, how can N contain all ordinals of M? Shouldn't a model contain ordinals as its domain? If models have same ordinals as its domain, aren't they basically the same, not needing any concept like inner model?
What am I getting wrong about a model? What makes an inner model different?
If I am not mistaken, it seems to say that ZFC has a countable inner model, which means countable domain - countable ordinals... and this just does not make sense.