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 $X\to X\times X$.)
The rule of thumb is this: Your DM (or Artin) stack will be a sheaf in the fppf/fpqc topology if the condition imposed on its diagonal is fppf/fpqc local on the target ("satisfies descent").
In other words, in condition 2 you asked that the diagonal be a relative scheme/relative algebraic space perhaps with some extra properties. If there if fppf descent for morphisms of this type (e.g., "relative algebraic space", "relative monomorphism of schemes"), you'll have something satisfying fppf descent. If there is fpqc descent for morphisms of this type (e.g., "relative quasi-affine scheme"), then you'll have something satisfying fpqc descent.
See for instance LMB (=Laumon, Moret-Bailly. Champs algebriques), Corollary 10.7. Alternatively: earlier this year I wrote up some notes (PDF link) that included an Appendix collecting in one place the equivalences of some standard definitions of stacks, including statements of the type above.
Best Answer
A canonical example of a sheaf of sets on a topological space $X$ is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$. The gluing property then says that a continuous function on a union of open subsets $U_i$ of $X$ is the same thing as a collection of continuous functions $f_i: U_i \to \mathbb{R}$ such that $f_i$ and $f_j$ coincide on $U_i∩U_j$.
A canonical example of an ∞-sheaf (alias stack) of groupoids on a topological space $X$ is the ∞-sheaf that sends an open subset $U$ of $X$ to the groupoid of finite-dimensional continuous real vector bundles on $U$. (Isomorphisms in this groupoid are continuous fiberwise linear isomorphisms of vector bundles on $U$.) The gluing property then says that a vector bundle on a union of open subsets $U_i$ of $X$ is the same thing as a collection of vector bundles $V_i$ on $U_i$, together with isomorphisms $t_{i,j}: V_i→V_j$ of vector bundles restricted to $U_i∩U_j$, and such that $t_{j,k}t_{i,j}=t_{i,k}$ on $U_i∩U_j∩U_k$. This last condition is known as the cocycle condition and in some textbooks vector bundles are defined in this manner.
So the point of triple intersections is that isomorphisms of vector bundles over pairwise intersections must themselves satisfy a higher coherence identity. This condition is trivial for sheaves of sets because two functions can be equal in exactly one way, unlike vector bundles, which can be isomorphic in many different ways.
To answer the second question: the analog of the etale space of a sheaf of sets is the etale stack of an ∞-sheaf of groupoids. This stack is no longer an ∞-sheaf on the original topological space, but rather on the site of all topological spaces. (Some care must be taken when dealing with size issues here, since topological spaces do not form a small category, but I suppress these issues here for simplicity.) This etale stack can be constructed in many different ways. For example, there is a unique homotopy cocontinuous functor from ∞-sheaves of groupoids on a topological space $X$ to ∞-sheaves of groupoids on all topological spaces that sends a representable sheaf given by an open subset $U$ of $X$ to the representable sheaf of $U$ as an object in the site of all topological spaces. The image of a given ∞-sheaf $F$ of groupoids under this functor $E$ is the etale stack $E(F)$ of $F$, which is equipped with a canonical morphism (in the category of ∞-sheaves of groupoids on topological spaces) to the representable sheaf of $X$.
If we now take the ∞-sheaf of sections of the resulting map $E(F)→X$ of stacks, we recover the original ∞-sheaf $F$.
(There are many other constructions, of course. For example, one could work instead with topological groupoids, or rather, localic groupoids, instead of sheaves of groupoids on topological spaces.)