Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly speaking $[X//G]$ is the fine moduli space for $M$; hence it carries a universal family and there's a correspondence between families of objects of $M$ over a scheme $S$ and maps $S\to [X//G]$. Now, I claim that also $X$ carries a universal family, obtained via pull-back from $[X//G]$, and it is constant along the fibers of the quotient map $X \to [X//G]$. 


If $S$ is once again a family of objects of $M$, what can one say of maps from $S\to X$? That the family over $S$ induces a map to $X$ that is unique up to $G$-action? That such a map is unique as long as $X\to [X//G]$ has a section? 


Basically, by the existence of the universal family over $X$ a map $S\to X$ induces a family over $S$ but not in a unique way. So, in the other direction, given a family over $S$ this induces an orbit under $G$ of maps, but there's no canonical choice of a particular map in this orbit.



universal families and maps to quotient stacks
First a couple corrections. If $M$ is represented by quotient stack, it better be a contravariant functor, and moreover, it should probably take values in groupoids, not set. Anyway, here's what going on:


$X$ does not carry a universal family, but it carries a "locally" universal family $v \in M\left(X\right)$, in the sense that, if $S$ is a scheme, and $x \in M\left(S\right)$ then you can find a cover (say in the etale topology, if that's where your stack is) $\left(f\_\alpha:S\_\alpha \to S\right)$ such that each $f\_\alpha^\*\left(x\right) \in M\left(S\_\alpha\right)$ arises, *up to isomorphism* as $g\_\alpha^\*\left(v\right)$ for some $g\_\alpha:S\_\alpha \to X.$ 


I haven't used the group $G$ yet. Lets do that. A morphism $S \to M$ corresponds to a choice of a $G$-torsor $P \to S$ and a $G$-equivariant map $P \to X.$ So the map $S \to M$ need not factor through $X,$ unless the torsor is trivial, but it always factors through etale-locally.


