What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is.

Now, in the contest of algebraic geometry, we refer to a moduli space as a scheme that (co)represents a particular functor, i.e. \begin{equation} \mathcal{M}:\,\{\mathfrak{schemes}\}^{\circ}\longrightarrow\{\mathfrak{sets}\} \end{equation} that is called the $moduli \,functor$. The question if it is (co)representable is called the $moduli \;problem$.

My question is, how does a differential geometer image a moduli space? That is, I thought that he thinks it in the same above way (replacing the category of schemes with a more suitable category), but it turns out that it is not so.

thank you!

EDIT: I have seen some people still look at this post, so maybe it is worth to add something. First of all, the geometry of a $C^{\infty}$-manifold is much less rigid than that one of an algebraic variety. This reflects, for example, on the fact that in general we can find a lot automorphisms of the objects we are interested in. In particular, most of the cases (and probably 'all' the cases) we have no hope to get a fine moduli space: this justify the fact that, in differential geometry, one usually talks about moduli space without specify if it is fine or not.

Secondly, regarding my description of the moduli problem. I have recently looked at the theory of $C^{\infty}$-schemes (see http://arxiv.org/pdf/1104.4951v2.pdf). This is a generalization of the concept of a $C^{\infty}$-manifold in the same way as a scheme is a generalization of an algebraic variety. An interesting aspect of this theory is, for example, that a $C^{\infty}$-manifold is always an affine $C^{\infty}$-scheme. Replacing the category of $C^{\infty}$-schemes in the moduli functor defined above, we can think at a moduli space of $C^{\infty}$-objects in the classical algebraic way. Moreover, this leads in a natural way to a theory of derived differential geometry.

Finally, I feel quite confident with this. Neverthless, this theory is modelled on the algebraic analogue and it is very recent. Even if my interests are principally on algebraic geometry, I had asked this question because I was wondering how mathematicians from different areas of mathematics approach to the problem of studying moduli spaces (the basic aim was to better understand the mathematical feeling behind these objects). And since moduli spaces in differential geometry were studied before (or at least without) the theory of $C^{\infty}$-schemes, my primary question is still not anwered.

Columbia University's 2013 Eilenberg Lecture series was delivered by Joe Harris, who opened it by remarking that it was not until Grothendieck that algebraic geometers used a formal definition of moduli spaces.

I am far from an expert. But when reading, for example, Hitchin or Donaldson I do not get the impression they are using a formal definition of moduli space in general. Rather, one calls particular constructions the "moduli spaces of X" e.g. the "moduli spaces Yang Mills instantons."

For example, connections on a principle $G$-bundle $\pi : E \to X$ form an affine space $\mathcal{A}$ and we quotient by the infinite-dimensional gauge group $\mathcal{G}$, worry about stability, and then call $\mathcal{A}/\mathcal{G}$ "the moduli space of connections moduli gauge" and prove facts about it. Notice we are starting with a moduli space rather than a moduli problem (for moduli functors in algebraic geometry, we start with the problem and say what counts as a solution).

Now a word about $C^\infty$-schemes. I suppose you could define moduli problems $F : C^\infty\text{-Sch} \to \text{Set}$ for solutions of a PDE, for Morse flow lines, etc. and this may buy you something. But, that is not exactly the point of $C^\infty$-schemes–their role is moduli theory is more subtle.

For some time, people like Kontsevich and Drinfeld prophesied of "hidden smoothness" or "intrinsic smoothness" in moduli spaces. That is, even when the moduli space sucks (badly singular, components of high dimension), there is a virtual fundamental class (in homology, in the Chow ring, etc.) of the virtual dimension (the virtual dimension is the dimension the moduli space would be if all the data was sufficiently generic). One can then define double secret spooky quantum counting invariants (Gromov-Witten, Donaldson-Thomas, Pandharipande-Thomas, etc.) by integrating against the virtual fundamental class as in $$\text{DT}^\alpha(\tau) = \int_{[\mathcal{M}^\text{ss}_\alpha(\tau)]^\text{vir}} 1.$$

In algebraic geometry, virtual fundamental classes are constructed using perfect obstruction theories (see the widely cited paper "The Instrinsic Normal Cone" by Behrend-Fantechi).

In differential geometry one should use "Kuranishi structures," but not everybody agrees on what the 'correct' definition should be. The original definition of Kuranishi structures was proposed by Fukaya-Oh-Onta-Ono in their work on the moduli space of J-holomorphic curves. Since then, Joyce, McDuff-Wehrheim, and Dingyu Yang have proposed alternative definitions. As I understand it, Joyce is using $C^\infty$-schemes and derived differential geometry to define Kuranishi structures (see https://arxiv.org/abs/1409.6908).

All relevant parties make their case with thousands of pages of technical material and saying anything intelligent about Kuranishi structures is way beyond my pay grade. And as a humble grad student, I do hope an actual expert will come along give you a better answer than I have (I am not even sure if I answered your question).