volume form
On an n-dimensional space there is one special degree of form: degree n, the top degree, which measures full n-dimensional volume. A volume form is a nowhere-zero n-form, and it is exactly the gadget that, fed n vectors, returns the signed volume of the box they span. In flat R^3 the standard volume form is dx^dy^dz; feed it three vectors and you get the determinant of the matrix with those vectors as columns — the signed volume you already know.
Two roles make the volume form indispensable. First, it is what you integrate a function against to get a total: the mass of a body with density rho is the integral of rho times the volume form, and in coordinates this is the familiar triple integral of rho dx dy dz. Second, the volume form is precisely what carries orientation: choosing a volume form is the same as choosing which way is 'positively oriented', and switching its sign swaps left-handed for right-handed. A space admits a global volume form exactly when it is orientable — which is why a Mobius band, being non-orientable, has no consistent volume form and you cannot integrate top-degree forms over it the usual way.
In curved space or curvilinear coordinates the volume form is not just dx^dy^dz; it picks up a factor from the metric. In polar coordinates it is r dr^d(theta); in spherical coordinates it is r^2 sin(phi) dr^d(theta)^d(phi); in general it is the square root of the determinant of the metric times the coordinate form. That factor is exactly the Jacobian scale factor you carry through any change of variables — so the volume form is the home of the change-of-variables determinant, and it is the object the divergence theorem and the whole machinery of integration on manifolds are built around.
In R^3 feed dx^dy^dz the three column vectors u = (1, 0, 0), v = (1, 2, 0), w = (1, 1, 3). The result is the determinant of [1, 1, 1; 0, 2, 1; 0, 0, 3] = 1*2*3 = 6, the signed volume of the parallelepiped they span. Reorder two of the vectors and the answer becomes -6 — orientation, recorded by the form.
The volume form returns the signed volume of the box its inputs span — a determinant that flips with orientation.
A volume form needs only orientation, not a metric — any nowhere-zero top-degree form will do, and two such forms differ by a positive function. But to get the geometric volume (lengths and angles as you expect), you need the metric volume form with its square-root-of-the-metric-determinant factor; bare dx^dy^dz gives the right volume only in Cartesian coordinates.