Suppose we have real $n$-vector bundle $p:E\to X$ over some $CW$-complex $X$. It is well-known that if $E$ is orientable (i.e. $\pi_1X$ acts on orientation of fibers trivially) then the obstruction to construct a nonzero section of $E$ is Euler class $e\in H^n(X,\mathbb Z)$.
In case $E$ is nonorientable one can define the orientation sheaf $\newcommand{\ZZ}{\widetilde{\mathbb Z}}$ $\ZZ$ (which fiber over a point $x$ is $H^{n-1}(p^{-1}(x)\setminus0,\mathbb Z)$), and then define the obstruction to construct a nonzero section of $E$ as certain element $\newcommand{\ee}{\widetilde{e}}$ $\ee\in H^n(X,\ZZ)$.
I have difficulties with computing $\ee$ in particular cases and with some properties of $\ee$. E.g. how to understend that $\ee(TM)=\chi(M)$ (with right orientation -- how to chose this orientation?). Is there any litrature describing this characteristic class and how to work with?