I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils down to:
"... which induces an exact sequence in homotopy $$\ldots\pi_2(S^1)\to\pi_2(S^3)\to\pi_2(S^2)\to\pi_1(S^1)\to\pi_1(S^3)\to\pi_1(S^1)\ldots$$ from which the higher homotopy groups of spheres could be computed ..."
Now I know exact sequences, some algebraic topology and how continuous maps induce maps on homology and how Mayer-Vietoris extends this (under certain conditions) to a long exact sequence, and how the Hurewicz theorem gets you from homology to homotopy, but the machinery behind the above statement seems to be some other result/theorem/piece of theory. A very crude guess is that $\pi_n$ is a functor which preserves exact sequences, and some kind of Mayer-Vietoris then gives the long exact sequence.
If someone has a nice reference for this material, that would be nice. To motivate you guys, here is a pretty picture.