I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and that an orientation with respect to real K-theory is a spin structure. I think this is a very beautiful picture and I am wondering if orientations with respect to other theories like elliptic cohomology, G-equivariant cohomology, quaternionic K-theory, or spin cobordism correspond to interesting and well-studied differential-geometric structures.

Complex manifolds should be oriented with respect to any complex-oriented cohomology theory. Indeed, if $E$ is a complex-oriented cohomology theory then all complex vector bundles carry $E$-orientations. In particular, if $X$ is a complex manifold then its tangent bundle $TX$ has a complex structure making $X$ an $E$-oriented manifold.

Given that complex cobordism is the universal complex-oriented cohomology theory, I would guess that an orientation with respect to complex cobordism is a complex structure. I have been unable to find any literature on this and I am unsure how to approach the problem rigorously on my own. Maybe someone knows?