This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level.

Unless I'm mistaken, the rough statement is that $H^n(X;G)$ (the $n$-dimensional cohomology of $X$ with coefficients in $G$) should somehow correspond to (free?) homotopy classes of maps $X \to K(G,n)$. I want to understand this better, in relatively elementary terms. Here are some questions which (I hope) will point me in the right direction.

1. What category are we working in? My guess is that $X$ should just be a topological space, the cohomology is singular cohomology, and our maps $X \to K(G,n)$ just need to be continuous.
2. Does this carry over if we give $X$ a smooth structure, take de Rham cohomology, and require our maps $X \to K(G,n)$ to be smooth?
3. How does addition in $H^n(X;G)$ carry over?
4. How does the ring structure on $H^*(X;G)$ carry over? (This has probably been adequately answered to Dinakar already.)