TITLE: Understanding the definition of the quotient stack $[X/G]$
QUESTION [14 upvotes]: 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.
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 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.

REPLY [6 votes]: It follows from the definitions that $[X/G]$ is representable by a scheme $S$ if and only if $X$ is a $G$-torsor over $S$, i.e. the natural map $G\times X\rightarrow X\times _SX$  given by $(g,x)\mapsto (x,gx)$ is an isomorphism. 
Example 0.4 (Chapter 0, §3, page 11 of 3rd edition) in Mumford's GIT is a variety $X$ with an action of $G=SL(2)$ with trivial stabilizers and geometric quotient $\mathbb{A}^1$, but such that $X$ is not a $G$-torsor over $\mathbb{A}^1$, so $[X/G]$ is not representable.