We define general characteristic classes and state and prove the Chern-Weil Theorem. We then introduce the Pontryagin classes of a vector bundle.

As an application, we show that $\mathbb{C}P^2$ does not embed in $\mathbb{R}^5$.

Of course this only scratches the surface of what is possible with characteristic classes! If you are interested, I recommend taking a course in Algebraic Topology.