a higher homotopy group
The fundamental group records how loops (maps of a circle) sit in a space. But why stop at circles? You can also ask how spheres of every dimension map into a space, up to deformation. The higher homotopy groups pi_n do exactly this: pi_n(X) measures the homotopy classes of maps of the n-sphere into X. They probe holes of every dimension and are, even now, among the deepest and least understood invariants in topology.
Fix a basepoint x_0 in X. Define pi_n(X, x_0) to be the set of homotopy classes (rel basepoint) of based maps f: (S^n, s_0) -> (X, x_0). Equivalently, use maps of the n-cube (I^n, boundary of I^n) -> (X, x_0) sending the cube's whole boundary to x_0. The group operation is built by stacking two cubes side by side along one coordinate and rescaling — the n-dimensional analogue of concatenating loops. For n = 1 this recovers pi_1. The striking new fact for n >= 2: pi_n is always abelian. The picture is the Eckmann-Hilton argument — with two or more free directions you can slide one sphere-map around the other, and the extra room forces the product to commute, something impossible with the single direction available to loops.
Higher homotopy groups are powerful but notoriously hard. pi_n(S^n) = Z (degree), matching intuition, but pi_3(S^2) = Z is already surprising (generated by the Hopf map, a nontrivial map of a 3-sphere into a 2-sphere), and the homotopy groups of spheres pi_{n+k}(S^n) form an intricate, still-incompletely-known pattern that is a central object of modern homotopy theory. Two honest cautions: unlike homology, higher homotopy groups are NOT computed by any simple van-Kampen-style gluing, which is much of why they are hard; and a space can have all higher pi_n vanish yet be far from contractible if pi_1 is nontrivial (e.g. the circle: pi_n(S^1) = 0 for n >= 2 but pi_1 = Z), so 'aspherical' (higher groups vanish) is weaker than 'contractible.'
The Hopf map S^3 -> S^2 generates pi_3(S^2) = Z. Build it by viewing S^3 as unit pairs (z, w) of complex numbers and S^2 as the Riemann sphere; send (z, w) to the ratio z / w. The preimage of each point is a circle, and distinct fibers link once — that linking is exactly the nontrivial class no deformation can remove.
The Hopf fibration shows pi_3(S^2) is nonzero — higher homotopy groups defy low-dimensional intuition.
pi_n for n >= 2 is abelian, but it still carries an action of pi_1 (the basepoint matters in non-simply-connected spaces), so 'pi_n(X)' is fully canonical only when X is simply connected; ignoring the pi_1-action is a common oversight.