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][1] starting on page 410.

More generally, if you started with an $(n-1)$-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 $\pi_n(X)$.


  [1]: http://www.math.cornell.edu/~hatcher/AT/ATchapters.html