As I understand it a standard model is a model in which the relation is the $\in$ on the actual set of sets constituting the model.
(i) Hence theories that aren't in the language of set $L_S$ generally won't have a standard model because the binary relation $\in$ can't model binary functions like $+$ in group theory for example. Is this right?
(ii) What do non-standard models of set theory look like? Would someone show me an example? If possible as simple as possible.
Many thanks for your help.