Why study characteristic classes?
A characteristic class is a way of associating to each principal bundle \(X\) a cohomology class of \(X\). The cohomology class measures the extent the bundle is “twisted” — and whether it possesses sections.
Characteristic numbers solve the oriented and unoriented bordism questions: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.
When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the Stiefel-Whitney class, the Chern class, and the Pontryagin classes) were reflections of the classical linear groups and their maximal torus structure.