I will explain how geometric descriptions of genera determine geometric descriptions of the associated cohomology theories and then give some examples. Then I will try to say something about the case of elliptic genera. For these the geometric description is still not rigorous.

References (I have copies of the non-web references, in case any one is interested):

1. Haven't looked at this paper but it has a cool title: Dijkgraaf, R.; Moore, G.; Verlinde, E.; Verlinde, H., Elliptic genera of symmetric products and second quantized strings. Comm. Math. Phys. 185 (1997), no. 1, 197--209. hep-th/9608096
2. Witten, Ed., Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (1987), no. 4, 525--536. Postscript from KEK library
3. Hopkins, Michael J. Characters and elliptic cohomology. Advances in homotopy theory (Cortona, 1988), 87--104, London Math. Soc. Lecture Note Ser., 139, Cambridge Univ. Press, Cambridge-New York, 1989
4. M. J. Hopkins, M. Ando, and N. P. Strickland, "Elliptic spectra, the Witten genus, and the theorem of the cube", dvi file
5. Segal, G. "Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)". Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162, (1988), Exp. No. 695, 4, 187--201 (1989).