I'm looking for cold hard facts about just what $BG$ classifies, if $G$ is any grouplike topological monoid. I have some vague idea that $[X,BG]$ is in bijection with equivalence classes of "principal fibrations" over $X$. What exactly is a principal fibration?


May's *Classifying Spaces and Fibrations* looks like a good source, but I'm having trouble teasing out whether his notion of $G\mathcal U$-fibration (which he proves is classified by $BG$) is equivalent to the notion of a fibration $E \rightarrow B$ with a fiberwise right $G$-action giving weak equivalences $G \rightarrow E\_b$, $g \mapsto xg$ for each point $x$ in the fiber $E\_b$. (I can show the $\Rightarrow$ but not the $\Leftarrow$. Maybe there needs to be another condition in my naive description.)


Also, any grouplike monoid $G$ is weakly equivalent to $\Omega BG$, so "principal fibrations," whatever they are, should correspond (in some sense I want to make precise) to pullbacks of the path-loop fibration over $BG$.



what does BG classify? i.e. what is a principal fibration?
In view of the references to my Memoir, Classifying spaces and fibrations,
in other answers, I guess I should answer too. The requested answer is
implicit but not quite explicit there. Fix a grouplike topological monoid 
$G$. Maybe assume for simplicity that its identity element is a nondegenerate 
basepoint (no loss of generality by 9.3). Define a $G$-torsor to be
a right $G$-space $X$ such that the $G$-map $G\longrightarrow X$ that sends 
$g$ to $xg$ is a weak equivalence for $x\in X$. Let $\mathcal{G}$ be the category of 
$G$-torsors and maps of $G$-spaces between them. The Memoir defines a
$\mathcal{G}$-fibration in terms of the $\mathcal{G}$-CHP, which is equivalent to having a $\mathcal{G}$-lifting function. Cary asks whether that notion is equivalent to an
a priori weaker notion. The answer depends on what ``equivalence'' means.
The Memoir insists on $\mathcal{G}$-fibrations, but it defines an equivalence (6.1) 
to be a $\mathcal{G}$-map over the base space, where a $\mathcal{G}$-map only has to be a map of $G$-torsors on fibers. (More precisely, it takes the equivalence relation generated by such maps.) Using $\mathcal{G}$-fibrations allows one often to replace 
that notion of equivalence by the nicer one of $\mathcal{G}$-fiber homotopy equivalence. But with the
equivalence relation as given, any $\mathcal{G}$-map that is a quasifibration is equivalent to a $\mathcal{G}$-fibration, by the $\Gamma$-construction in Section 5. 
Therefore the classification theorem remains true allowing all $\mathcal{G}$-spaces that are quasifibrations, which of course includes Cary's preferred notion of a 
$G$-fibration. 


