An alternative description of the first Stiefel-Whitney class I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please?
The first Stiefel-Whitney class of a vector bundle is an element in the first cohomology group of the base space $H^1(B,Z_2)$. This element can be seen as a map $w_1 : \pi_1(B) \rightarrow Z_2$
(because of the universal coefficient theorem and the fact that $H_1$ is the abelianization of $\pi_1$), so it assigns to each loop based at a fixed point an element of $Z_2$. Now this number is $0$ if and only if that loop is orientation preserving (locally we have an orientation and when we go around a loop and come back to our first place, we can ask if orientation has changed or not)

If you have a triangulated manifold then it is orientable when its n simplices can be given signs (+/-1) so that the signed sum is an integer cycle.
If the manifold is not orientable then any signed sum of the n simplices is not a cycle and it turns out that what goes wrong is that some of the n-1 simplices in the boundary get counted twice rather than cancelling each other out.
This boundary which is a sum of doubles of n-1 simplices can be divided by two to give a 2 torsion Z homology class. This n-1 dimensional cycle represents the obstruction to orienting the manifold. I would strongly suspect that the first Stiefel Whitney class of the tangent bundle is the Poincare dual to this homology class. Check it out. 
