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An orientation on a smooth manifold is a consistent choice of "handedness" at every point — a continuous selection of one of two equivalence classes of ordered bases for the tangent space. Equivalently, it is a nowhere-vanishing top-degree differential form (a volume form). Not every manifold is orientable: the Mobius band and the Klein bottle are classic non-orientable examples. Orientation is necessary for integration of differential forms to yield a well-defined signed quantity.
At each point p of an n-manifold M, the tangent space TpM is an n-dimensional vector space. An ordered basis (v₁, ..., vₙ) for TpM is called positively oriented or negatively oriented relative to a reference basis — two ordered bases have the same orientation if the change-of-basis matrix has positive determinant. This divides all bases into two equivalence classes. An orientation on M is a smooth (continuous) choice of one class at each point.
The differential-forms perspective makes this cleaner. An n-form ω on an n-manifold is a smooth section of the top exterior power Λⁿ(T*M). At each point, the space of n-forms is one-dimensional, so ω_p is either positive, negative, or zero (relative to a basis). A volume form is a nowhere-vanishing n-form — it picks out a "positive" orientation at every point. The manifold is orientable if and only if a volume form exists. In coordinates, a volume form looks like f(x) dx¹ ∧ ... ∧ dxⁿ where f > 0 everywhere (in positively oriented charts).
The prototypical non-orientable surface is the Mobius band: take a rectangle and glue two opposite edges with a twist. Walking around the band, your notion of "clockwise" flips by the time you return to the start. No continuous assignment of clockwise/counterclockwise is possible. The Klein bottle (a closed non-orientable surface) and the real projective plane ℝP² are other fundamental examples. For the projective plane, non-orientability follows because the antipodal map on S² reverses orientation (it has degree -1 in even dimensions).
Orientation is not just a topological curiosity — it is essential for integration. The integral of an n-form over an oriented n-manifold is well-defined: you break the manifold into coordinate patches, integrate in each patch, and sum via partition of unity. The orientation ensures that overlapping patches contribute consistently (transition maps have positive Jacobian determinant). Reversing the orientation flips the sign of the integral. On non-orientable manifolds, you can still integrate densities (which transform by |det J| rather than det J), but forms themselves cannot be integrated consistently. Stokes' theorem requires an orientation because the boundary must be compatibly oriented with the interior.