The 
th
Stiefel-Whitney class of a real vector
bundle (or tangent bundle or a real manifold) is in the 
th cohomology group of the base space
involved. It is an obstruction to the existence of
real linearly independent vector
fields on that vector bundle, where 
is the dimension of the fiber. Here,
obstruction means that the 
th Stiefel-Whitney class being nonzero
implies that there do not exist 
everywhere linearly independent vector
fields (although the Stiefel-Whitney classes are not always the obstruction).
In particular, the 
th Stiefel-Whitney class is the obstruction to the existence
of an everywhere nonzero vector
field, and the first Stiefel-Whitney class of a manifold
is the obstruction to orientability.
