How does one define a standard model of ZFC?

Question

I sometimes come across the phrase, "a standard model of ZFC." Is this a rigorous concept? If so, how does one define it?

Answer 1

A standard model of ZFC is a pair $(M,E)$ such that $M$ is non-empty and $E$ is well-founded, and $(M,E)\models\sf ZFC$. Using Mostowski's collapse lemma this is isomorphic to a transitive set, $N$ such that $(N,\in\upharpoonright_N)\models\sf ZFC$.

Standard models are exactly those models of ZFC whose ordinals are well-ordered (externally), and if they are transitive then their ordinals are exactly some ordinal $\alpha$.