Are the non-standard models of classical analysis the standard models of non-standard analysis? I've never quite understood how the non-standard models of classical analysis related to the standard models of non-standard analysis. I know that non-standard analysis got its start with Robinson (and others) studying non-standard models of the reals, but I don't know enough about non-standard analysis to know whether the field developed beyond that. 
Are the non-standard models of classical analysis all and only the standard models of non-standard analysis?
 A: Firstly, nonstandard analysis developed far beyond the (quite a lot of) theory the Robinson developed. 
There is no standrad model for nonstandard analysis. Arguably, there never will be, but regardless, and more importantly, there probably never could or should be.
The standard model of analysis exists because the second order axioms of complete real ordered fields are categorical. Every two models are isomorphic, thus there is essentially just one model - voila a standard one. But of course there is no standard one, just several standard constructions of the reals, which nobody cares about since you work with any model axiomatically anyways. 
For nonstandard analysis there is no such categoricity. Nonstandard models, as developed by Robinson, exists because of the compactness theorem. Other constructions exists, for instance by using ultraproducts. In any case, there is no explicit construction of any nonstandard model. Every construction relies on some rather strong principle of choice, and thus we can't ever have a very explicit description of a nonstandard model. 
Ultimately, the goal of nonstandard analysis is to do analysis. Very standard analysis. The aim is to develop proof tools and techniques that are nonstandard in order to better understand analysis. So, it is not so much any nonstandard model that we wish to study, but rather the extension of the reals to an nonstandard model + the transfer principle. 
The transfer principle says that even though an extension of a standrad model to a nonstandard one (no matter which one you choose) is quite wild and is mostly a blackbox, the first order theory of both models is strongly related. Basically, every first order sentence (with bounded quantification) holds in the standard model iff its $^*$ transform holds in the extended model. With that in place, the particularities of the extension are, to a large extend, irrelevant. 
So, to summarize, there is very little hope of there ever being anything remotely like an explicit construction of a nonstandard model, let alone one that shall deserve to be called 'standard'. Secondly, even if there were, it is the transfer principle that we care about, not so much the extension itself. 
I hope this helps.  
