the orientation of a manifold
An orientation is a consistent choice of 'which way is positive' across a whole manifold. On a line it is a direction; on a surface it is a consistent sense of clockwise-versus-counterclockwise, or equivalently a continuously chosen 'up' side; in higher dimensions it is a consistent right-hand-rule for ordering basis vectors. The trouble is that not every manifold admits such a global choice — the Möbius band famously does not, because walking once around flips your sense of handedness.
Precisely, an orientation of an n-manifold M is a smooth choice of orientation (an equivalence class of ordered bases, two being equivalent if the change-of-basis matrix has positive determinant) on each tangent space T_p M, varying continuously with p. Three equivalent ways to package this: (i) an atlas whose chart transitions all have positive Jacobian determinant; (ii) a nowhere-vanishing top-degree form omega in Omega^n(M), called a volume form or orientation form; (iii) a consistent choice of generator of the top exterior power Lambda^n(T*_p M) at each point. A manifold is orientable if such a choice exists, and then there are exactly two orientations on each connected component.
Orientation is the prerequisite for integrating forms: the integral of a top-degree form over M is only defined once M is oriented, because reversing orientation flips the sign of the integral (just as reversing the direction of a definite integral flips its sign). It is therefore baked into Stokes' theorem, where the boundary inherits an induced orientation. An honest caveat: orientability is a topological property independent of any metric — RP^2 and the Klein bottle are non-orientable, the sphere and the torus are orientable. And do not conflate orientability (existence of a choice) with an orientation (a specific choice); a connected orientable manifold has two.
The sphere S^2 is orientable: the outward unit normal gives a consistent 'outside', equivalently the area form (restricted from R^3) is a nowhere-zero 2-form. The Möbius band is NOT orientable: try to choose a consistent normal and after one loop around the core circle it points the opposite way, so no nowhere-vanishing top form exists. Concretely, on R^n the standard orientation is the one for which dx^1 ^ dx^2 ^ ... ^ dx^n is positive — the right-handed ordering of axes.
Orientability is global: locally every manifold has two orientations, but a Möbius band cannot glue them into one consistent choice.
Orientation needs no metric — it is pure smooth topology, captured by a nowhere-vanishing top form. A non-orientable manifold has no such form and so admits no honest integration of top-degree forms (one must use densities instead). The two notions, orientable and oriented, are different: the first is yes/no, the second is a chosen sign.