Differential Forms

Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

Concrete

Differential forms (co-vectors) are functions (elements of dual vector-space) which map vectors to real numbers.

For the basic idea, see http://jakobschwichtenberg.com/vectors-forms-p-vectors-p-forms-and-tensors/
One of the best introductions can be found in “Geometrical methods of mathematical physics” by Bernard F. Schutz
Manifolds and Differential Forms lecture notes by Reyer Sjamaar
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard - Extremely student friendly, lots of margin notes that talk about the "soft" stuff that's so crucial to the actual practice of math. Reading just the margins jumps your mathematical maturity by 2 years.
Another good introduction can be found in “Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts”(2021) by Tristan Needham

Researcher

The motto in this section is: the higher the level of abstraction, the better.

Why is it interesting?

P-forms are important, because they are exactly the objects that we need if we want to talk about areas and volumes (and higher dimensional analogues).

http://jakobschwichtenberg.com/vectors-forms-p-vectors-p-forms-and-tensors/

‘Hamiltonian mechanics cannot be understood without differential forms’.

Mathematical methods of classical mechanics by Wladimir Igorewitsch Arnold, p. 163