differential form
Think of all the things you actually integrate in vector calculus. Sometimes you integrate a function over a region (a double or triple integral). Sometimes you integrate F dot dr along a curve. Sometimes you integrate a flux F dot n dA across a surface. These look like different operations with different notations, yet they are all the same kind of act: feeding a little oriented piece of space into a gadget and reading off a number. A differential form is exactly that gadget — the thing you put under an integral sign.
More precisely, a differential form of degree k (a k-form) is an object that, at each point of space, eats k little tangent vectors (the edges of a tiny oriented k-dimensional parallelepiped) and returns a number, and it does so antisymmetrically — swap two of the input vectors and the answer flips sign. A 0-form is just a scalar function f, which eats no vectors and returns its value. A 1-form like 3 dx + x dy eats one vector and returns a number; you have already met its integral as a line integral. A 2-form eats two vectors (an oriented area patch) and is what you integrate over a surface. The notations dx, dy, dz are not informal symbols to cancel — they are the basic 1-forms, the coordinate-measuring slots, and you build everything else from them.
Forms are the right language because they make orientation, dimension, and the change-of-variables Jacobian automatic rather than bolted on by hand. They unify gradient, curl, and divergence into one operator (the exterior derivative d), and they turn Green's, Stokes', and the divergence theorems into a single statement (the generalized Stokes' theorem). They are the working notation of differential geometry, electromagnetism written in spacetime, Hamiltonian mechanics, and modern applied analysis on curved or higher-dimensional spaces.
In R^3 the general 1-form is P dx + Q dy + R dz, the general 2-form is A dy^dz + B dz^dx + C dx^dy, the general 3-form is f dx^dy^dz, and a 0-form is just f. Integrating the 1-form P dx + Q dy + R dz along a curve is exactly the line integral of the vector field (P, Q, R), so the form is a repackaging of objects you already know.
In 3-space there are forms of degree 0, 1, 2, 3 only — beyond degree equal to the dimension every form is zero.
A form is not the same as a vector field, even though both have several components in R^3. The translation between them depends on a choice of metric and orientation; in curved space or under a non-rigid change of coordinates the form transforms correctly on its own, while a naive vector field does not.