Just like there is a universal cover of every space, there is a natural n$n$-connected space X_n$X_n$ that maps to any space X$X$. To construct this space, you can add cells of dimension n+2$n+2$ and higher to X$X$ to get a space Y$Y$ together with a map X \to Y$X \to Y$ which is an isomorphism on \pi_i$\pi_i$ for i \leq n$i \leq n$, but such that \pi_i(Y)=0$\pi_i(Y)=0$ for i>n$i>n$. The homotopy fiber X_n \to X$X_n \to X$ of this map is then the "n"$n$-connected cover" of X; X_n$X$; $X_n$ is n$n$-connected but has the same homotopy groups as X$X$ above n$n$, as can easily be seen from the long exact sequence of the fibration. DetailsDetails of this, as well as a proof of uniqueness of the n$n$-connected cover, are in Hatcher starting on page 410.
More generally, if you started with an (n-1)$(n-1)$-connected space, you could both kill the homotopy groups of X$X$ above n$n$ and kill a subgroup of \pi_n(X)$\pi_n(X)$, and then the homotopy fiber would be an "n"$n$-cover" of X$X$ corresponding to that subgroup of \pi_n(X)$\pi_n(X)$.
