A real line bundle is a rank-one real vector bundle. Over a paracompact Hausdorff space, or over a smooth manifold in the smooth category, it is orientable exactly when it is a trivial vector bundle.
To prove the nontrivial implication, choose a fiber metric. An orientation of a vector bundle selects one of the two unit vectors in each fiber. These choices agree continuously in oriented local frames, giving a global nowhere-zero section. The map is then a fiber-linear product isomorphism. Conversely, a trivial vector bundle has the positive standard basis as a continuous orientation of a vector bundle.
The Möbius line bundle over a circle is the basic counterexample to triviality: one circuit reverses the fiber orientation, so no global positive unit section of a vector bundle can be chosen.
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