In Spivak Calculus on Manifolds the author uses a definition of orientation of a manifold which I do not understand, and which I do not found elsewhere. I cite:
It is often necessary to choose an orientation $\mu_x$ for each tangent space $M_x$ of a manifold $M$. Such choices are called consistent provided that for every coordinate system $f : W \to \mathbb R^n$ and $a, b \in W$ the relation $$ [f_{\ast}((e_1)_a), \ldots, f_{\ast}((e_k))_a] = \mu_{f(a)} $$ holds if and only if $$ [f_{\ast}((e_1)_b), \ldots, f_{\ast}((e_k)_b)] = \mu_{f(b)}. $$
For the notation. With $v_p$ he denotes the tangent vector $v$ at the point $p$, for a function $f : A \to \mathbb R^m$ with $A \subseteq \mathbb R^n$ differentiable on $A$ we have $f_{\ast}(v_p) := (Df(p)(v))_{f(p)}$ and for a basis $v_1, \ldots, v_n$ he denotes by $[v_1, \ldots, v_n]$ the orientation of $v_1, \ldots, v_n$. Also a coordinate system is defined as:
A subset $M$ of $\mathbb R^n$ is a $k$-dimensional manifold if and only if for each point $x \in M$ the following ''coordinate condition'' is satisfied:
(C) There is an open set $U$ containting $x$, an open set $W \subseteq R^k$, and a $1-1$ differentiable function $f : W \to \mathbb R^n$ such that
(1) $f(W) = M \cap U$,
(2) $f'(y)$ has rank $k$ for each $y \in W$,
(3) $f^{-1} : f(W) \to W$ is continuous.
Such a function $f$ is called a coordinate system around $x$.
Now my question, I totally do not understand his definition of orientability of a manifold. If I got him right he requires that we find a number $\mu_x$ for each point $x$ on the manifold, which fulfills a compatibility condition for every coordinate system. As $\det Df(p) \ne 0$ for each $p \in W$, I see that this determinant could not switch sign, but otherwise it could vary continuously. So it does not make sense to have a numbers $\mu_x$ and require the above condition for all coordinate systems, as by composition with diffeomorphisms we can change the coordinate system and take care that the determinant might have different values (despite still having all the same sign). So this might be an error and instead ''for all coordinate'' systems, he means ''for each $x$ there exists a coordinate system such that'' (i.e. we have an atlas whose transition maps have determinants with the same sign, which is closer to other definitions I found). Another possiblity is how $[\cdot,\cdots,\cdot]$ is precisely defined, as he just wrote it denotes the orientation (or equivalence class), but such an class is not a number, its a set of ordered bases, but could be represented by a number, so $[\cdot,\cdots,\cdot] \in \mathbb R$, but maybe we have $[\cdot,\cdots,\cdot]\in\{-1,1\}$ which would make some more sense.
Other definitions I found are that the existence of a $n$-form $\omega$ on $M$ is asserted such that $\omega(p)$ is strictly positive on the tangent space at $p$, or that we could find an atlas such that the transition maps have positive determinant, or the definitions from here which I hardly understand.
(1) On ${\bf R}^n$ fix $n$-form $\omega=dx_1\cdots dx_n$ Then at each point $x\in {\bf R}^n$ we have basis $\{e_i\}$. Then $$ \omega_x (e_1,\cdots, e_n)>0\ {\rm or}\ <0 $$
Hence we have two classes, and we choose first class as orientation at $x$, denoted by $[e_1,\cdots, e_n]_x$
That is if $e_i$ is a vector field on ${\bf R}^n$ as we already know, $[e_1\cdots e_n]$ determine orientation at each point.
(2) If $f: W\subset M \rightarrow {\bf R}^n$ is a coordinate chart, then orientation consistent to $f$ on $W$ is given by $$ [df^{-1} e_1 \cdots df^{-1} e_n ]$$
If $E_i$ is basis on $T_yM$, and if $ \omega (df E_i,\cdots,df E_n) >0 $ then $[E_1\cdots E_n]_y=[df^{-1} e_1 \cdots df^{-1} e_n ]_y $, i.e., $[E_1\cdots E_n]_y$ represented an orientation at $y$.
(3) If $g : U\rightarrow {\bf R}^n$ is another chart and $U\cap W\neq \emptyset $, then a manifold is orientable iff for any $f,\ g$, $$ [df^{-1} e_1\cdots df^{-1} e_n]_y=[dg^{-1} e_1\cdots dg^{-1} e_n]_y $$ for $y\in U\cap W$.