Stacks and sheaves
Let me see if I understand your example correctly: you are fixing $X$ and $Y$, families of curves over $S$, and now you are considering the functor which maps an $S$-scheme $T$ to the set of $T$-isomorphisms $f^*X \to f^*Y$ (where $f$ is the map from $T$ to $S$).
If I have things straight, then this functor shouldn't be so bad to think about, because it is actually representable, by an Isom scheme. In other words, there is an $S$-scheme $Isom_S(X,Y)$ whose $T$-valued points, for any $f:T \to S$, are precisely the $T$-isomorphisms from $f^*X$ to $f^*Y$. (One can construct the Isom scheme by looking inside a certain well-chosen Hilbert scheme.)
One way to think about this geometrically is as follows: one can imagine that two curves over $k$ (a field) are isomorphic precisely when certain invariants coincide (e.g. for elliptic curves, the $j$-invariant). (Of course this is a simplification, and the whole point of the theory of moduli spaces/schemes/stacks is to make it precise, but it is a helpful intuition.) Now if we have a family $X$ over $S$, these invariants vary over $S$ to give a collections of functions on $S$ (e.g. a function $j$ in the genus $1$ case), and similarly with $Y$. Now $X$ and $Y$ will have isomorphic fibres precisely at those points where the invariants coincide, so if we look at the subscheme $Z$ of $S$ defined by the coincidence of the invariants, we expect that $f^*X$ and $f^*Y$ will be isomorphic precisely if the map $f$ factors through $Z$. Thus $Z$ is a rough approximation to the Isom scheme.
It is not precisely the Isom scheme, because curves sometimes have non-trivial automorphisms, and so even if we know that $X_s$ and $Y_s$ are isomorphic for some $s \in S$, they may be isomorphic in more than one way. So actually the Isom scheme will be some kind of (possibly ramified) finite cover of $Z$.
Of course, if one pursues this line of intuition much more seriously, one will recover the notions of moduli stack, coarse moduli space, and so on.
Added: The following additional remark might help:
The families $X$ and $Y$ over $S$ correspond to a map $\phi:S \to {\mathcal M}_g \times {\mathcal M}_g$. The stack which maps a $T$-scheme to $Isom_T(f^*X, f^*Y)$ can then seen to be the fibre product of the map $\phi$ and the diagonal $\Delta:{\mathcal M}_g \to {\mathcal M}_g \times {\mathcal M}_g$.
In the particular case of ${\mathcal M}_g$ the fact that this fibre product is representable is part of the condition that ${\mathcal M}_g$ be an algebraic stack.
But in general, the construction you describe is the construction of a fibre product with the diagonal. This might help with the geometric picture, and make the relationship to Mike's answer clearer. (For the latter:note that the path space into $X$ has a natural projection to $X\times X$ (take the two endpoints), and the loop space is the fibre product of the path space with the diagonal $X\to X\times X$.)
I'm not a geometer, but here's one way to think of it. Let M be a stack (in groupoids, say) and X an object, and consider just a single point p:X→M. Then $Hom_M(p,p)$ is a sheaf, which is the "isotropy group" of the stack M at the point p, i.e. the space of automorphisms of p in M. If you think of your stacks as like topological spaces, with isomorphisms corresponding to paths (which makes the most sense when you move up to ∞-stacks, where higher morphisms correspond to higher homotopies between paths), then the isotropy-group object of a point p corresponds to the loop space ΩX of a topological space X.
As far as the first part of your question goes, I have exactly the same confusion (and it will probably get worse once I start thinking about sheaves on stacks...).
There are two things that give me the illusion of some comprehension.
- Thinking of sheaves as (generalized) spaces is probably not terrible, in the same way you think of a bundle just as its total space. It is indeed true that any sheaf (with values in a reasonable category I guess) is the sheaf of sections of its etale space. (Although I must confess I don't particularly like etale spaces and I'm aware that this is probably not the right picture, as we shouldn't think of sheaves on a site in this fashion, but hey).
- If we think of fibred categories (+cleavage) as presheaves with values in the 2-category Cat, then there is a generalization of the sheaf condition which yields the notion of a stack. Of course, arrows being a sheaf is a consequence of this, so if you find the generalization of the sheaf condition more natural, then arrows-being-a-sheaf might be viewed as a formal consequence I guess.
Anyway it's just a thought.