The Stiefel-Whitney number is defined in terms of the Stiefel-Whitney class of a manifold as follows. For any collection of Stiefel-Whitney classes such that their cup product has the same dimension as the manifold, this cup product can be evaluated on the manifold's fundamental class. The resulting number is called the Pontryagin number for that combination of Pontryagin classes.
The most important aspect of Stiefel-Whitney numbers is that they are bordism invariant. Together, Pontryagin and Stiefel-Whitney numbers determine an oriented manifold's bordism class.
