If $E \to X$ is an oriented rank $k$ vector bundle, then $E$ has an Euler class $e(E) \in H^k(X; \mathbb{Z})$ which is the first obstruction to the existence of a nowhere-zero section of $E$.
If $E$ is non-orientable, then I have heard that $E$ has a twisted Euler class $e(E) \in H^k(X; \mathbb{Z}_w)$; here $H^k(X; \mathbb{Z}_w)$ denotes cohomology with local coefficients where $\mathbb{Z}_w$ is the $\mathbb{Z}[\pi_1(X)]$-module given by the homomorphism $w : \pi_1(X) \to \operatorname{Aut}(\mathbb{Z})$ corresponding to $w_1(E)$ under the isomorphism of groups $\operatorname{Hom}(\pi_1(X), \operatorname{Aut}(\mathbb{Z})) \cong \operatorname{Hom}(\pi_1(X), \mathbb{Z}_2) \cong H^1(X; \mathbb{Z}_2)$.
Does anyone know of a reference for the construction of the twisted Euler class? In particular, I'd like a reference which shows that it is the first obstruction to the existence of a nowhere-zero section.
If $M$ is a closed smooth non-orientable $n$-dimensional manifold, then we have $e(TM) \in H^n(M; \mathbb{Z}_w)$. We also have a twisted fundamental class $[M] \in H_n(M; \mathbb{Z}_w)$.
I'd also like a reference for the claim $\langle e(TM), [M]\rangle = \chi(M)$.