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This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.

Let's say we begin with a naive moduli problem: we want a moduli space (whatever space would mean) that would "classify" all gadgets.

The first, obvious attempt, is to set $F$ as a functor from $Schemes$ to $Sets$ like so: $F(S):=$ all the families of gadgets parametrized by $S$. Sometimes $F$ will not be representable by a scheme.

Now we may ask more cleverly, is there a stack fibered in groupoids given by $F(S):=$ all families of gadgets parametrized by $S$, with isomorphisms corresponding the various $S$-automorphisms.

It seems that people think of stacks as the "natural" object for moduli problems, but I do occasionally see papers proving the existence of a moduli stack. So let me phrase it like this:

The failure for the moduli space to be a scheme has to do with non-trivial automorphisms. What does the failure of the moduli space to be a stack indicative of? What does the existence of a moduli stack imply?

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Being a stack roughly means it is a sheaf valued in categories (or more specifically, groupoids). So if the moduli problem is not a stack it probably means that we don't always have effective descent for objects or morphisms, i.e. we cannot glue things sometimes. Maybe you mean algebraic stacks, which is not quite far away from schemes. –  shenghao Nov 25 '10 at 18:16
The point of being an alg. stack is that it has enough "geometric" structure that it is meaningful to define connectedness, dimension, cohomology, etc. for $F$. Proving $F$ is a "stack fibered in groupoids" is usually a triviality (once one is familiar with descent theory) and is not so interesting by itself, whereas having the further property of being an algebraic stack requires real work. A lesson of Artin's work is that if there is a rich enough deformation theory for $F$, then approximation methods create a "scheme chart" to provide geometric structure on $F$. That's the real content. –  BCnrd Nov 25 '10 at 19:37
To give an example, define $F(S)$ to be category of elliptic curves over $S$, with isoms as morphisms (and usual pullback functoriality in $S$). By the theory of (families of) elliptic curves, $F$ is a "stack fibered in groupoids". Snore. The real beef in making a "scheme chart" (to prove $F$ is a Deligne-Mumford stack) is: find an elliptic curve $E \rightarrow M$ so that the resulting map $M \rightarrow F$ is "etale". Loosely speaking, means that any family of elliptic curves becomes isomorphic to pullback of $E \rightarrow M$ etale-locally on the base. That lies deeper than making up defns. –  BCnrd Nov 25 '10 at 19:46
Thanks, Brian! - –  James D. Taylor Nov 25 '10 at 20:27
Small (but important) correction: in my 2nd comment, I should have said that the requirement on $E \rightarrow M$ is that the resulting map $M \rightarrow F$ is an "etale cover", not just "etale". (The covering aspect is crucial, like with charts on a manifold. This feature was implicit in the "Loosely speaking" interpretation I gave for what it means.) –  BCnrd Nov 26 '10 at 3:41
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