Any coordinate patch in a manifold is orientable. Indeed, the image is orientable — we can just use to orient — and given a choice of top form on we can pull it back along to give an orientation of itself.
But what happens when we bring two patches and together? They may each have orientations given by top forms and . We must ask whether they are “compatible” on their overlap. And compatibility means each one picks out the same end of at each point. But this just means that — when restricted to the intersection — for some everywhere-positive smooth function .
Another way to look at the same thing is to let be the pullback , and . Then we must ask what this function is. It must exist even if the orientations are incompatible, since is never zero, but what is it?
A little thought gives us our answer: is the Jacobian determinant of the coordinate transformation from one patch to the other. Indeed, we use the Jacobian to change bases on the cotangent bundle, and transforming between these top forms amounts to taking the determinant of the transformation between the -forms and .
So what does this mean? It tells us that if the Jacobian of the coordinate transformation relating two coordinate patches is everywhere positive, then the coordinates have compatible orientations. On the other hand, if the coordinate transformation’s Jacobian is everywhere negative, then the coordinates also have compatible orientations. Why? because even though the sample orientations differ, we can just use and , which do give the same orientation everywhere.
The problem comes up when the Jacobian is sometimes positive and sometimes negative. Now, it can never be zero, but if the intersection has more than one component it may be positive on one and negative on the other. Then if you pick orientations which coincide on one part of the overlap, they must necessarily disagree on the other part, and no coherent orientation can be chosen for the whole manifold.
I won’t go into this example in full detail yet, but this is essentially what happens with the famous Möbius strip: glue two strips of paper together at one end and we can coherently orient their union. But if we give a half-twist to the other ends before gluing them, we cannot coherently orient the result. The Jacobian is positive on the one overlap and negative on the other.