Under a not unreasonable assumption about cardinal arithmetic, namely $2^{<c}=c$ (which follows from the continuum hypothesis, or Martin's Axiom, or the cardinal characteristic equation t=c), the number of non-isomorphic possibilities for *R of cardinality c is exactly 2^c. To see this, the first step is to deduce, from $2^{<c} = c$, that there is a family X of 2^c functions from R to R such that any two of them agree at strictly fewer than c places. (Proof: Consider the complete binary tree of height (the initial ordinal of cardinality) c. By assumption, it has only c nodes, so label the nodes by real numbers in a one-to-one fashion. Then each of the 2^c paths through the tree determines a function f:c \to R, and any two of these functions agree only at those ordinals $\alpha\in c$ below the level where the associated paths branch apart. Compose with your favorite bijection R\to c and you get the claimed maps g:R \to R.) Now consider any non-standard model *R of R (where, as in the question, R is viewed as a structure with all possible functions and predicates) of cardinality c, and consider any element z in *R. If we apply to z all the functions *g for g in X, we get what appear to be 2^c elements of *R. But *R was assumed to have cardinality only c, so lots of these elements must coincide. That is, we have some (in fact many) g and g' in X such that *g(z) = *g'(z). We arranged X so that, in R, g and g' agree only on a set A of size $<c$, and now we have (by elementarity) that z is in *A. It follows that the 1-type realized by z, i.e., the set of all subsets B of R such that z is in *B, is completely determined by the following information: A and the collection of subsets B of A such that z is in *B. The number of possibilities for A is $c^{<c} = 2^{<c} = c$ by our cardinal arithmetic assumption, and for each A there are only c possibilities for B and therefore only 2^c possibilities for the type of z. The same goes for the n-types realized by n-tuples of elements of *R; there are only 2^c n-types for any finite n. (Proof for n-types: Either repeat the preceding argument for n-tuples, or use that the structures have pairing functions so you can reduce n-types to 1-types.) Finally, since any *R of size c is isomorphic to one with universe c, its isomorphism type is determined if we know, for each finite tuple (of which there are c), the type that it realizes (of which there are 2^c), so the number of non-isomorphic models is at most (2^c)^c = 2^c.
To get from "at most" to "exactly" it suffices to observe that (1) every non-principal ultrafilter U on the set N of natural numbers produces a *R of the desired sort as an ultrapower, (2) that two such ultrapowers are isomorphic if and only if the ultrafilters producing them are isomorphic (via a permutation of N), and (3) that there are 2^c non-isomorphic ultrafilters on N.
If we drop the assumption that $2^{<c}=c$, then I don't have a complete answer, but here's some partial information. Let \kappa be the first cardinal with 2^\kappa > c; so we're now considering the situation where \kappa < c. For each element z of any *R as above, let m(z) be the smallest cardinal of any set A of reals with z in *A. The argument above generalizes to show that m(z) is never \kappa and that if m(z) is always < \kappa then we get the same number 2^c of possibilities for *R as above. The difficulty is that m(z) might now be strictly larger than \kappa. In this case, the 1-type realized by z would amount to an ultrafilter U on m(z) > \kappa such that its image, under any map m(z) \to \kappa, concentrates on a set of size < \kappa. Furthermore, U could not be regular (i.e., (\omega,m(z))-regular in the sense defined by Keisler long ago). It is (I believe) known that either of these properties of U implies the existence of inner models with large cardinals (but I don't remember how large). If all this is right, then it would not be possible to prove the consistency, relative to only ZFC, of the existence of more than 2^c non-isomorphic *R's.
Finally, Joel asked about a structure theory for such *R's. Quite generally, without constraining the cardinality of *R to be only c, one can describe such models as direct limits of ultrapowers of R with respect to ultrafilters on R. The embeddings involved in such a direct system are the elementary embeddings given by Rudin-Keisler order relations between the ultrafilters. (For the large cardinal folks here: This is just like what happens in the "ultrapowers" with respect to extenders, except that here we don't have any well-foundedness.) And this last paragraph has nothing particularly to do with R; the analog holds for elementary extensions of any structure of the form (S, all predicates and functions on S) for any set S.
A difference between what Gel'fand did and what the Germans were doing is that in 1930s-style algebraic geometry you had the basic geometric spaces of interest in front of you at the start. Gel'fand, on the other hand, was starting with suitable classes of rings (like commutative Banach algebras) and had to create an associated abstract space on which the ring could be viewed as a ring of functions. And he was very successful in pursuing this idea. For comparison, the Wikipedia reference on schemes says Krull had some early (forgotten?) ideas about spaces of prime ideals, but gave up on them because he didn't have a clear motivation. At least Gel'fand's work showed that the concept of an abstract space of ideals on which a ring becomes a ring of functions was something you could really get mileage out of. It might not have had an enormous influence in algebraic geometry, but it was a basic successful example of the direction from rings to spaces (rather than the other way around) that the leading French algebraic geometers were all aware of.
There is an article by Dieudonne on the history of algebraic geometry in Amer. Math. Monthly 79 (1972), 827--866 (see http://www.jstor.org/stable/pdfplus/2317664.pdf) in which he writes nothing about the work of Gelfand.
There is an article by Kolmogorov in 1951 about Gel'fand's work (for which he was getting the Stalin prize -- whoo hoo!) in which he writes about the task of finding a space on which a ring can be realized as a ring of functions, and while he writes about algebra he says nothing about algebraic geometry. (See http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=6872&what=fullt&option_lang=rus, but it's in Russian.) An article by Fomin, Kolmogorov, Shilov, and Vishik marking Gel'fand's 50th birthday (see http://www.mathnet.ru/php/getFT.phtmljrnid=rm&paperid=6872&what=fullt&option_lang=rus, more Russian) also says nothing about algebraic geometry.
Is it conceivable Gel'fand did his work without knowing of the role of maximal ideals as points in algebraic geometry? Sure. First of all, the school around Kolmogorov didn't have interests in algebraic geometry. Second of all, Gel'fand's work on commutative Banach algebras had a specific goal that presumably focused his attention on maximal ideals: find a shorter proof of a theorem of Wiener on nonvanishing Fourier series. (Look at http://mat.iitm.ac.in/home/shk/public_html/wiener1.pdf, which is not in Russian. :)) A nonvanishing function is a unit in a ring of functions, and algebraically the units are the elements lying outside any maximal ideal. He probably obtained the idea that a maximal ideal in a ring of functions should be the functions vanishing at one point from some concrete examples.
Best Answer
Try this: www.dpmms.cam.ac.uk/~cb496/nsag1.pdf and also this http://wwwmath.uni-muenster.de/u/serpe/documents/ultramath2008serpe-nonstandard-handout.pdf, logicandanalysis.org/index.php/jla/article/view/77/29 and references therein.