This question arose after I thought about Ben Webster's comments to this question.

There he asked me what was my definition of a moduli problem. When I came to think of it, I never saw a precise definition like that. My understanding is along the following lines.

Roughly, say, we want to describe the moduli problem classifying objects of a certain type. In the functorial formulation, we would have a functor

$$Schemes \rightarrow Sets $$ $$ X \mapsto \{ Iso.\ Classes\ of\ some\ objects\ of\ a\ certain\ type\ defined\ over\ X.\}$$

And if this functor is representable, we say that a fine moduli space exists for this moduli problem, and even if it is not, if a certain one-one correspondence between points and objects is true over algebraically closed fields, and if a certain universal property for this is satisfied, then a coarse moduli space exists.

I suppose the above is the agreed standard terminology. Please correct me if I am wrong.

Now, the problem is that in the above definition of a moduli problem, the notion of a "functor classifying isomorphism classes of a certain type of object" is vague. We could have curves with marked points, other types of varieties with extra conditions, bundles, and so on. If we on the other hand relax the criteria and allow just any functor, then the definition becomes too broad,and any scheme will be a fine moduli space for its functor of points.

So is there a better definition, or is this all one can say? Pardon me if this was a stupid question.