I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's *Algebraic Stacks and Moduli of Vector Bundles*. 

Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The **quotient stack** $[X/G]$ is the pseudofunctor
$$
[X/G]:(\mathsf{Sch}/S)^{\operatorname{op}}\to\mathsf{Grpds}
$$
defined as follows. 

For an $S$-scheme $U$ let $[X/G](U)$ be the category whose objects are diagrams
$$
U\xleftarrow{\pi}E\xrightarrow{\alpha}X
$$
where $\pi$ is a principal $G$-bundle and $\alpha$ is a $G$-equivariant morphism. The morphisms in $[X/G](U)$ are the isomorphisms of principal $G$-bundles commuting with the $G$-equivariant morphisms.

For a morphism of $S$-schemes $f:U^\prime\to U$ let
$$
[X/G](f):[X/G](U)\to[X/G](U^\prime)
$$
be the functor induced by pullbacks of principal $G$-bundles.

Now, it is clear to me that $[X/G]$ is isomorphic to the representable functor $\operatorname{Hom}(-,X)$ when $G$ is the trivial group. It is not clear to me how to determine when $[X/G]$ is representable in general. My questions are:

**Question 1.** Is there a sufficient condition we can impose on $\rho$ to ensure that $[X/G]$ is representable? The [Wikipedia article][1] on quotient stacks says something about when the categorical quotient $X/G$ exists the canonical map $[X/G]\to\operatorname{Hom}(-,X/G)$ need not necessarily be an isomorhpism. This is somewhat opaque to me as I'm not even sure how $[X/G]\to\operatorname{Hom}(-,X/G)$ is defined.

**Question 2.** What is an example where $[X/G]$ is *not* representable?

If these questions are too broad, I'd be very grateful if someone could point me in the direction of a good reference.


  [1]: http://en.wikipedia.org/wiki/Quotient_stack