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 1D loops, and not spheres.
There's certainly a homotopytheoretic 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). 


Just like there is a universal cover of every space, there is a natural $n$connected space $X_n$ that maps to any space $X$. To construct this space, you can add cells of dimension $n+2$ and higher to $X$ to get a space $Y$ together with a map $X \to Y$ which is an isomorphism on $\pi_i$ for $i \leq n$, but such that $\pi_i(Y)=0$ for $i>n$. The homotopy fiber $X_n \to X$ of this map is then the "$n$connected cover" of $X$; $X_n$ is $n$connected but has the same homotopy groups as $X$ above $n$, as can easily be seen from the long exact sequence of the fibration. Details of this, as well as a proof of uniqueness of the $n$connected cover, are in Hatcher starting on page 410. More generally, if you started with an $(n1)$connected space, you could both kill the homotopy groups of $X$ above $n$ and kill a subgroup of $\pi_n(X)$, and then the homotopy fiber would be an "$n$cover" of $X$ corresponding to that subgroup of $\pi_n(X)$. 


There may be a geometric but partial answer to your question. This is an idea I learnt from Dennis Sullivan. As we know, passing from $X$ to its universal cover $\widetilde{X}$ kills the fundamental group. Now, by Hurewicz we can assume that $H_2(\widetilde{X})=\pi_2(\widetilde{X})$, whence killing $H_2(\widetilde{X})$ suffices. If we assume $H_2(\widetilde{X})$ is torsion free then each generator $\alpha_i\in H_2(\widetilde{X})$ corresponds to a circle bundle $E_i$ over $\widetilde{X}$, i.e., $H_2(E_i)=H_2(\widetilde{X})/\mathbb{Z}\alpha_i$. Thus, if $\widetilde{X}$ was a manifold of dimension $n$ and $H_2(\widetilde{X})$ was free of rank $k$ then taking successive circle bundles we get a manifold $E$ of dimension $n+k$. This has the same higher homotopy groups ($\pi_i$ for $i>2$) as that of $X$. The example given by Reid Barton is an illustration of this. However, for manifolds this is as far as you can go since killing even the free part of $\pi_3(\widetilde{X})$ (or, equivalently the free part of $H_3(E)$) requires bundles over $E$ with fibre $\mathbb{CP}^\infty$, which lands us outside the realm of finite dimensional manifolds. 


The previous answers gave analogues to the universal covering space and looked at the homotopy groups of these analogues. However, the analogy to the $n=1$ case is not complete: While $\pi_1(X)$ classifies the automorphisms over $X$ of the universal covering space, the $\pi_n(X)$ don't classify the automorphisms over $X$ of these $n$connected analogues. In fact, the higher $\pi_n(X)$ don't seem to classify anything else than homotopy classes of maps $S^n \to X$. This is one of the motivations for using $n$groupoids as invariants of spaces, see the discussion here, right before the references: http://ncatlab.org/nlab/show/fundamental+group+of+a+topos or, starting on page 17 with a nice story on the invention of the higher homotopy groups, and the desire for a nonabelian homology: http://www.intlpress.com/hha/v1/n1/a1/ 


Rather than just taking homotopy groups for a single dimension, you can also think about the kind of algebraic entity that detects homotopy in two consecutive dimensions, or indeed any number of consecutive dimensions. In this discussion, following on from an earlier post, we're looking at the fundamental 2group which picks up homotopy in dimensions 1 and 2. 

