Before diving into concrete mathematical details, I was wondering if anyone could explain to me the essence of de Rham cohomolohy and why does it pop up all over (fundamental) mathematical physics.
Please regard this as simply a (probably misguided) attempt at an answer, because I wanted to learn a little bit more myself about the use of forms.
More information can be found at Wikipedia De Rahm Cohomology .
An analogy regarding the essence of De Rahm cohomology.
Assume there is an electron, hidden somewhere in space. You can find it only by using integrals that are based around physical, measurable properties of the object. So which integrals would be of use?
It's an electron, so using surface integrals along with Guass's law and you find a value for the electric flux. Then you can repeat this process until you get a non zero answer. This allows you to narrow the search to a given spherical shaped region of space. You narrow it down by incrementally reducing the size of the sphere.
The point of all this is to try to demonstrate the usefulness of de Rham cohomology as a general procedure for finding objects in a space that possess a given type of field. The field describes differential forms. You assume that the field is strong enough, and that its strength goes to infinity as you get closer to the object, so that it causes a defect in space. You want to locate this defect, by computing certain integrals and checking that they are not zero. Looking at whether this form integrates to zero or not on various hypersurfaces of the appropriate dimension corresponds to figuring out what the differential form looks like in de Rham cohomology.
Another Intuitive Approach
Homology and cohomology are, amongst other things, a way of counting the number of holes in a manifold.
Assume a 2-D plane with a point missing out of it. Now, if you have any two points anywhere, given the definition of topology, you can squash them together into one "bigger" point. Then just go around and avoid the missing point. However, if you have a string looped around the missing point, you can't squish that loop to a point. You end up with a 1-dimensional hole.
Or, for another example, consider a torus. There are some one-dimensional, string loops on the torus you can bring to a point (aka "null-homotopic closed paths"), but there are some you can't. For instance, any one of these loops cannot be brought to a point:
Not alone are you unable to bring these loops together, you can't even deform the red one to become the purple one. Those are two distinct 1-dimensional holes in our space/manifold, so the 1-D homology (or cohomology) is going to have two independent generators in this situation.
Any shape inside the space is a hole if it has no boundary or it is not the boundary of anything else.
This stresses the difference between closed loops and open paths; it also says that a closed loop is not a hole if it bounds a patch. A surface with a boundary is not thought being a hole, but a surface without boundary (like a sphere) is, and it's a hole unless it has a full sphere inside it.
0-forms are just functions $f(x,y,z)$
1-forms are like $fdx+gdy+hdz$
2-forms are like $fdxdy+gdxdz+hdydz$
Forms can be differentiated and integrated. Differentiating a 0-form gives a 1-form, a 1-form gives a 2-form, and so on.
In physics, these options are equivalent to "grad", "div" and "curl".
A form is closed if its derivative is 0, and a form is exact if it is the derivative of something else. This should be seen as analogous to the two parts of "being a hole" dealt with above: being closed is like "having no boundary", and being exact is like "bounding a patch". A hole is a closed form which is not exact.
If I'm not mistaken, it's a valuable tool because it encodes global topological properties of some space in an algebraic object, which should be easier to work with, right? Where would one struggle without it?
Speaking in physical terms, finding a form whose derivative is a given form is like finding a "potential". It is found by integration, and it is vital that you can define those integrals in a unique way, completely independent of the path of integration. This is the main reason why the differential structure is an alternative way of viewing the geometric structure of paths and holes.
This is a summary of De Rahm Cohomology.