TITLE: Turning cobordism into a cohomology theory
QUESTION [16 upvotes]: 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.

REPLY [4 votes]: The OP asked about a description of the cohomology theory of Bordism, which no one above did in an explicit way.
Given a manifold $X$, an element of $\Omega^*(X)$ is represented by a smooth proper map $M \xrightarrow{i} X$ together with the data of the normal bundle $\nu(i)$ of $i$.  $(M_1,i_1)$ and $(M_2,i_2)$ are equivalent if there is a (smooth proper) map $M \xrightarrow{i} X \times I$ such that $\nu(i)$ restricts to $\nu(i_1)$ and $\nu(i_2)$ at the $\partial M =M_1 \sqcup M_2$.
This cohomology class will be in degree $deg(X)-deg(M_1)$.  For example when $X$ is a point, one gets cohomology in the negative degrees.  This corresponds to the fact that for any spectrum $E$-homology of a point will be $\pi_*(E)$ and the $E$-cohomology of a point will be $\pi_{-*}(E)$.  Its easy to see from this explicit construction that $MO^*(pt)$ is exactly $MO_{-*}(pt)$ as it should be.  Checking that this is the cohomology theory dual to bordism homology in the sense of Akhil Matthew's answer can by done by looking at the details of the Pontryagin Thom construction.
For more information see the paper by Quillen: Elementary Proofs Of Some Results Of Cobordism Theory Using
Steenrod Operations where this was introduced.