In Michael Stone and Paul Goldbart 's book mathematics for physics, the orientable manifold is defined as follow:
A manifold or surface is orientable if we can choose a global orientation for the tangent bundle. The simplest way to do this would be to find a smoothly varying set of basis-vector fields, $e_µ(x)$, on the surface and define the orientation by choosing an order, $e_1(x), e_2(x),\cdots, e_d(x)$, in which to write them. In general, however, a globally defined smooth basis will not exist (try to construct one for the two-sphere, $S^2$ !). We will, however, be able to find a continuously varying orientated basis $e_1^{(i)}(x), e_2^{(i)}(x),\cdots, e_d^{(i)}(x)$ for each member, labelled by (i), of an atlas of coordinate charts. We should choose the charts so that the intersection of any pair forms a connected set. Assuming that this has been done, the orientation of a pair of overlapping charts is said to coincide if the determinant, det A, of the map $e_µ^{(i)} = A_µ^νe_ν^{(j)}$ relating the bases in the region of overlap, is positive. If bases can be chosen so that all overlap determinants are positive, the manifold is orientable and the selected bases define the orientation.
I don't understand what does the "globally defined smooth basis" mean. Does it mean as the sketch below? Where we start from some point say South Pole with locally defined basis. As we go along a longitude, we arrive at the North Pole with another locally defined basis. But if we go down along another longitude, we have a different locally defined basis in South Pole, so the basis is not globally smooth. But these two locally defined South Pole basis can be transformed to each other with positive determinant.
What they mean is that, for certain cases of manifolds (we call "parallelizable"), you can find a set of vector fields which are defined everywhere on your manifold, and form a basis of the tangent space at each point. And when such a global basis exists (it's called a global frame), you can define an orientation by just ordering this basis.
The thing is, for a general manifold, that's not always true. You can always find such a basis locally (because locally, your manifold looks like a copy of $\mathbb{R}^n$, and you can find such a basis in $\mathbb{R}^n$), but globally, it might not exist: not every manifold is parallelizable.
They give the example of $S^2$ because there is a famous theorem (the Hairy ball theorem) which says that any vector field on an even-dimensional sphere has to vanish at least once.
So in particular, you are not gonna be able to find a set of functions $e_1, e_2$ on $S^2$ such that for every $x \in S^2$, $e_1(x), e_2(x)$ form a basis of $T_{x}S^2$. Because by the above theorem, $e_1$ and $e_2$ will each vanish somewhere.