I've recently finished one semester in differential topology (with Milnor's Topology from the Differentiable Viewpoint) and my first semester of algebraic topology. I believe I understand Milnor's definition of what it means for two manifolds to be cobordant: roughly speaking, two n-manifolds are cobordant if there exists an (n+1)-manifold whose boundary is the disjoint union of the two original n-manifolds.
My question is how does one go from this definition to a cohomology theory? I'm not exactly sure this is the right question to ask, so please provide any insight you can. I naively understand spectra and Brown representability, so you can answer in those terms if you'd like (I'm sure these concepts have something to do with the answer, but I can't quite piece it together). The wiki page for complex cobordism may have the answer, but again, I can't decipher it.