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