The connection between the fundamental group and covering spaces is quite fundamental.  Is there any analogue for higher homotopy groups?  It doesn't make sense to me that one could make a branched cover over a set of codimension 3, since I guess, my intuition is all about 1-D loops, and not spheres.
There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ for $n \geq 2$. We could next ask for a $2$-connected cover $X''$ of $X'$: a space $X''$ with $\pi_kX = 0$ for $k \leq 2$ and a map $X'' \to X'$ which is an isomorphism on $\pi_n$ for $n \geq 3$. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension $1$ - it will be a $K(\pi_2X, 1)$. (For the universal cover the fiber was the discrete space $\pi_1X = K(\pi_1X, 0)$.)

An example is the Hopf fibration $K(\mathbb{Z}, 1) = S^1 \to S^3 \to S^2$.

Geometrically it's harder to see what's going on with the $2$-connected cover than with the universal cover, because fibrations with fiber of the form $K(G, 1)$ are harder to describe than fibrations with discrete fibers (covering spaces).
