Real (or complex) cobordism is described by a symmetric ring spectrum MO (or MU respectively) as explained in examples 2.8 and 2.9 here. Associated to such a ring spectrum $R$, we have a unit spectrum $GL_1(R)$ (see for example chapter 22 in the book by May and Sigurdsson.

Is there a nice description or interpretation of the units $GL_1(MO)$ or $GL_1(MU)$?

Let me "define" what I mean by nice by giving an analogy. For the units of $K$-theory, we can identify $GL_1(KU) = BU_{\otimes} \times \mathbb{Z}/2\mathbb{Z}$ and we know that this is the classifying space for virtual vector bundles of virtual dimension $\pm 1$, similarly for $KO$.

`$U\to GL_1(S)$`

(resp.`$O \t GL_1(S)$`

) and the description of MU explicitly makes the composite`$U\to GL_1(S) \to GL_1(MU)$`

nullhomotopic as a map of infinite loop spaces (similarly for $MO$). Another is that even though $MO$ is equivalent to a product of Eilenberg-Mac Lane objects,`$GL_1(MO)$`

is (as an infinite loop space) not.`$GL_1$`

is tough, and a good chunk of literature in orientation theory (much by Peter May) is devoted to determining information about it. – Tyler Lawson Mar 28 '13 at 12:41