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.
On
First I should say that I am not a logician so I hope I'm not being totally wrong here. I'm not entirely sure what you mean by the first questions. For the second one, suppose that $M$ is your favourite model for, say, $ZF$. Let's assume that $M$ is standard, so that the elements in $M$ are actual sets and the interpretation of $\in$ is the actual element-hood relation. Let us further assume that $M$ is countable (this is not really needed but you'll see why I add it). Let $f: \mathbb N \to M$ be a bijection.
Define a new model for $ZF$ on $M'=\mathbb N$. Thus, the elements in $M'$ are the 'sets'. Interpret $n\in m$ iff $f(n)\in f(m)$. Clearly, $M'$ is a model of $ZF$ and in it $\in$ is not the actual element-hood for members in $\mathbb N$.
Of course, if $M$ were not countable then you can still do the same trick and re-interpret the model using a bijection between $M$ and any appropriate set of your liking.
On
Three quick remarks, rather than a proper answer
1) We certainly talk of standard models of theories other than set theory -- e.g. we talk of standard and non-standard models of first-order Peano arithmetic $PA$. There is an intended interpretation of its language $L_A$, but other ways of interpreting its non-logical vocabulary which still make the axioms of $PA$ come out true.
2) Going right back to Skolem, we know that there must be non-standard countable models of set theory. But there is a limit to how much we can say about them in describing "what they look like". For if I could come up with a nicely detailed story constructing the model and showing that it is a model, I could presumably regiment it in set theory, thereby proving the consistency of ZFC inside ZFC which we know is impossible.
3) Haim Gaifman's paper on the idea of non-standard models is well worth reading: http://www.columbia.edu/~hg17/nonstandard-02-16-04-cls.pdf
(1) A standard model of set theory is one in which the membership relation is $\in$; one can have standard models of other theories (e.g., $\Bbb N$ for Peano arithmetic) that don’t say anything about membership.
(2) Let $\varphi(x)$ be $\exists y\big(x=\big\langle y,0\rangle\big)$, and let $\Phi$ be the class of $x$ satisfying $\varphi(x)$. For $x=\big\langle y,0\big\rangle$ and $u=\big\langle v,0\big\rangle$ in $\Phi$ let $x\,E\,u$ iff $y\in v$. More formally, $E$ is the class of $\langle x,u\rangle$ satisfying the formula $\psi(x,u)$ given by
$$\exists y\exists v\left(x=\big\langle y,0\big\rangle\land u=\big\langle v,0\big\rangle\land y\in v\right)\;.$$
Then $\langle\Phi,E\rangle$ is a non-standard class model of set theory that mimics $\langle\mathbf{V},\in\rangle$ in the obvious way. For example, you can check that for $x,u\in\Phi$ as above we have $(x\subseteq u)^\Phi$ iff $y\subseteq v$.