A differential form is a generalisation of the notion of a differential that is independent of the choice of coordinate system. An n-form is an object that can be integrated over an n-dimensional domain, and is the wedge product of n differential elements. For example, f(x) dx is a 1-form in 1 dimension, f(x,y) dx ∧ dy is a 2-form in 2 dimensions (an area element), and f(x,y,z) dx ∧ dy + g(x,y,z) dx ^ dz + h(x,y,z) dy ∧ dz is a 2-form in 2 dimensions (a surface element).
As dx ∧ dx = 0, an n form in k dimensions will have independent elements. For example, in three dimensions, a function of 0-forms will be a scalar field, a function of 1-forms will be a covector field (and therefore dual to a vector field), a function of 2-forms will be a bivector or pseudovector field, and a function of 3-forms has only one independent element and can be associated with a scalar field.
The derivative operation on an n-form is an n+1-form; this operation is known as the exterior derivative. By the generalized Stokes' theorem, the integral of a function over the boundary of a manifold is equal to the integral of its exterior derivative on the manifold itself.