canonical map of a monoid to its classifying space 
*

*Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.]

*Every small category $C$ has a classifying space $BC$, defined as the geometric realization of the nerve. [Martin Brandenburg.]

*The classifying space $BM$ of a monoid $M$ is (by definition) the classifying space  of the corresponding category.[Martin Brandenburg.]


In Homology Fibrations and the "Group-Completion" 
Theorem. 
McDuff, D.; Segal, G., line 2, "there is a canonical map of $H$-spaces from $M\to \Omega BM$ from $M$ to the space of loops on $BM$". What is this canonical map?
 A: Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The geometric realization $BM=|N_\bullet(M)|$ of this simplicial space is the classifying space $BM$. (This includes the discrete case.)
Unveiling the definitions, it is given by $$BM=(\coprod\limits_{n\ge0}M^n\times\Delta^n)/\tilde{}.$$
Now for each element $m\in M$, 
Now for each element $m\in M$, one gets an obvious map $$\Delta^1\rightarrow M\times\Delta^1\rightarrow BM,$$ which factors over $S^1\cong\Delta^1/\partial\Delta^1,$ so one gets a loop in $BM$. Putting all those together yields the canonical map $M\rightarrow\Omega BM$.
This is the same thing both for the standard and the so-called fat geometric realization of simplicial spaces. The latter one is the one used in the McDuff-Segal paper. 

Edit: RSQ asked for a more detailed description of the "obvious map". I'll try to be as detailed as possible.
Recall the definition of the standard n simplex $$\Delta^n=\{(x_0,...,x_n)\in\mathbb{R}^{n+1}|\sum_{i=0}^nx_i=1, 0\le x_i\le1\}$$ Let $m\in M$ and define $$l_m\colon \Delta^1\rightarrow BM$$ as the composition of $$\Delta^1\rightarrow M\times\Delta^1,(t_0,t_1)\mapsto(m,t_0,t_1)$$ the inclusion $$M\times\Delta^1\rightarrow \coprod\limits_{n\ge0}M^n\times\Delta^n$$ and the quotient map $$\coprod\limits_{n\ge0}M^n\times\Delta^n\rightarrow BM=(\coprod\limits_{n\ge0}M^n\times\Delta^n)/\tilde{}.$$ By precomposing with the homeomorphism $$[0,1]\rightarrow \Delta^1,t\mapsto(t,(1-t))$$ we get a path $$[0,1]\rightarrow BM$$ which I'll call $l_m$, too. Now we want to see that this is actually a loop: It holds $$l_m(0)=[m,(0,1)]=[e,0]=[m,(1,0)]=l_m(1)$$ where $[\_]$ denotes equivalence classes in $BM$ and the second and the third equality follows from the relations $\tilde{}$ we imposed on $BM$ (You can find them in any text about simplicial sets or simplicial spaces.). It follows that $l_m$ factors as $$S^1=[0,1]/\{0,1\}\rightarrow BM,$$ which I'll call $l_m$, too. Now finally define $$M\rightarrow \Omega BM, m\mapsto l_m.$$
