Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*mathbb{C}$P$^{infty}$ or a statement about Thom classes. It seems that later in Hopkins notes he says that the complex orientations of E are in one to one correspondence with multiplicative maps $MU rightarrow E$, is there a treatment that starts with this perspective? How do the complex orientations of a spectrum E help one compute the homotopy of $E$, or the $E$-(co)homology of MU? Further, what other kinds of orientations could we think about, are there interesting $ko$ or $KO$ orienations? how much of these $E$-orientations of X is detected by E-cohomology of X?

I do have some of the key references already in my library, for example the notes of Hopkins from '99, Rezk's 512 notes, Ravenel, and Lurie's recent course notes. If there are other references that would be great. I am secretly hoping to get some insight from some of the experts. (I guess I should really also go through Tyler's abelian varieties paper)

(sorry for the on and off texing but the preview is giving me weird feedback.)

EDIT: I eventually found the type of answer i was looking for in some notes of Mark Behrens on a course he taught. This answer is that a ring spectrum $R$ is complex orientable is there is a map of ring spectra $MU to R$. This also appears in COCTALOS by Hopkins but neither source takes this as the more fundamental concept. Anyway, the below answer is more interesting geometrically.