Introduction to the Geometry of Foliations, Part $A$, Authors Gilbert Hector and Ulrich Hirsch, Page $15$.
Let $\mathcal{F}$ be a Foliation on a Manifold $M$ defined by the atlas $\{(U_i,\phi_i)\}$.
The Foliation $\mathcal{F}$ is called Orientable if there is a subatlas of $\{(U_i,\phi_i)\}$ such that all $\phi_{ij}$ (the tansition maps) coming from this subatlas are orientation preserving.
What do they mean by saying "the tansition maps are orientation preserving" (keep in mind, these maps are not necessarily linear)?
For a regular foliation $\mathcal{F}$ of codimension $q$ on a $n-$dimensional manifold $M$ the transition maps are of the form \begin{equation*} \phi_{ij}(x,y)=(\phi^1_{ij}(x,y),\phi^2_{ij}(y))\subset \mathbb{R}^{n-q}\times \mathbb{R}^q \end{equation*} for $x\in U\subset \mathbb{R}^{n-q}$ and $y\in O\subset \mathbb{R}^q$ some open sets. The foliation being orientable means that you can find an atlas such that all the functions \begin{equation*} \mathrm{det}((\frac{\partial \phi^1_{ij}}{\partial x_k})_{1\le k\le n-q}) \end{equation*} are positive. Another, coordinate free way of saying that a foliation is orientable is to say that the vector bundle \begin{equation*} \wedge^{n-q}T\mathcal{F} \end{equation*} is trivializable.