Here is the easiest generalization of the fact you cite about Eilenberg-MacLane spaces. Spaces $X$ with exactly two nontrivial homotopy groups $\pi_n(X), \pi_m(X), 2 \le n < m$ are classified (up to (weak) homotopy equivalence) by these two homotopy groups together with one additional *Postnikov invariant*, which is a cohomology class in $H^{m+1}(B^n \pi_n(X), \pi_m(X))$, where $B^n A$ denotes the $n$-fold delooping $K(A, n)$ of a discrete abelian group $A$. This class classifies the fibration

$$B^m \pi_m(X) \to X \to B^n \pi_n(X).$$

If $n = 1$ then we need the additional data of the action of $\pi_1(X)$ on $\pi_m(X)$, and then the cohomology above is group cohomology with nontrivial / local coefficients.  

*Example.* The $3$-truncation of $S^2$ has two nontrivial homotopy groups $\pi_2(X) \cong \pi_3(X) \cong \mathbb{Z}$, and all other homotopy groups vanish. The Postnikov invariant is a class in $H^4(B^2 \mathbb{Z}, \mathbb{Z}) \cong H^4(\mathbb{CP}^{\infty}, \mathbb{Z}) \cong \mathbb{Z}$, and I believe it turns out to be a generator. 

The natural generalization of this fact is the theory of <a href="http://ncatlab.org/nlab/show/Postnikov+system">Postnikov towers</a>, although there the relevant Postnikov invariants are defined in terms of spaces defined in terms of other Postnikov invariants, so it's trickier to tell whether the're different or the same.