For example, concerning de Rham cohomology as explained on the lower part of page 5 here, one considers the vector space of all forms on a manifold $M$, $A_{DR}(M)$ in the notation of the paper (?).

The $p$-th de Rham cohomology is then defined as the quotient of the kernel and the image of the exterior derivative $d$ acting on the space of $p$ and $p-1$ forms respectively as

\[H_{DR}^p(M) = \frac{Ker(d: A_{DR}^{p}(M)\rightarrow A_{DR}^{p+1}(M))} {Im(d: A_{DR}^{p-1}(M)\rightarrow A_{DR}^{p}(M))}\]

Can cohomology always be defined as the kernel divided by the image of a certain operation?

What does cohomology mean or "measure" in as simple as possible intuitive terms in the de Rham case but also more generally?

PS: I tried to read wikipedia of course, but it was (not yet?) very enlightening for me ...