A Gadget defined for Complex Vector Bundles. The Chern classes of a Complex Manifold are the Chern classes of its Tangent Bundle. The th Chern class is an Obstruction to the existence of everywhere Complex linearly independent Vector Fields on that Vector Bundle. The th Chern class is in the th cohomology group of the base Space.
See also Obstruction, Pontryagin Class, Stiefel-Whitney Class