Hatcher: Barycentric Subdivision At the bottom of page 122 of Hatcher, he defines a map $S:C_{n}(X)\rightarrow C_{n}(X)$ by $S\sigma=\sigma_{\#}S\Delta^{n}$. What is the $S$ on the right hand side and how does it act on the simplex $\Delta$? I'm having trouble deconstructing the notation here.

It's the $S$ he defines in (2) of his proof: barycentric subdivision of linear chains. 
I think you will benefit from reading the following supplementary notes.
update: here's the idea: first he deals with linear chains so he knows how to define the map $S: LC_n(\Delta^n) \to LC_n(\Delta^n)$, i.e. he knows how to find the barycentric subdivision of the standard $n$-simplex (thought of as the identity map onto itself). Then he uses the map $\sigma_\#$ composed with the subdivided standard $n$-simplex to define a barycentric subdivision of singular $n$-chains.
