a differential form
A differential form is the thing you integrate. A 1-form is what sits under an integral sign in a line integral (like f dx + g dy), a 2-form is what you integrate over a surface (a flux), and a top-degree form is what you integrate over the whole manifold to get a volume. The point is that forms are built to be integrated: they already carry the orientation and the change-of-variables behaviour that makes the integral independent of coordinates.
Precisely, a differential k-form on a smooth manifold M is a smooth assignment of an alternating k-tensor omega_p on the tangent space T_p M to each point p — that is, a smooth section of the exterior bundle Lambda^k(T*M). In a coordinate chart (x^1, ..., x^n) it is written omega = sum over increasing multi-indices I = (i_1 < ... < i_k) of f_I dx^{i_1} ^ ... ^ dx^{i_k}, where the f_I are smooth functions. A 0-form is just a smooth function. The space of all k-forms is denoted Omega^k(M), and Omega*(M) = direct sum of all Omega^k(M) is a graded-commutative algebra under the wedge product.
Forms are the central objects of this whole field: the exterior derivative d, the pullback, the interior product, the Lie derivative, integration, and Stokes' theorem are all operations on Omega*(M). They are dual to vector fields and beautifully coordinate-free — a k-form can be evaluated on k vector fields to give a function, with no metric required. A common confusion: a k-form is NOT a covariant k-tensor field in general; it is the alternating (antisymmetric) ones only. The metric tensor g is a symmetric 2-tensor and is not a 2-form.
On R^2 the 1-form omega = -y dx + x dy, when integrated around a loop, gives twice the enclosed signed area; its value on a tangent vector v = (a, b) at the point (x, y) is -y a + x b. On R^3 the 2-form F = E_1 dy^dz + E_2 dz^dx + E_3 dx^dy packages a flux: integrating it over a surface gives the flux of the field (E_1, E_2, E_3) through that surface. Both are smooth sections of an exterior bundle.
1-forms integrate over curves, 2-forms over surfaces, n-forms over the whole n-manifold — degree matches dimension of the domain of integration.
Do not confuse a 1-form with a vector field. They look alike in coordinates (both are n-tuples of functions) but transform oppositely under a change of coordinates: a vector field uses the Jacobian, a 1-form its inverse-transpose. Without a metric there is no canonical way to turn one into the other.