I have never studied topology (other than point-set topology). I am now taking a course on manifolds using Spivak volume 1. I am having some difficulty understanding the motivation for defining de Rham cohomology, as expected probably, since I have no idea what "cohomology" is. The definition seems (to me) to be arbitrary and strange (quotient space of closed by exact forms). It isn't at all clear to me what knowing the de Rham cohomology of a manifold is supposed to tell me about the manifold. I gather that (perhaps some of) the motivation for making this definition is that this space is isomorphic to some topological construction which I don't understand. My professor assures me that I needn't understand cohomology to understand de Rham cohomology, so I am asking here, what is the motivation for defining de Rham cohomology?