group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Where tmf is obtained from the moduli stack of elliptic curves, $tmf_0(2)$ (or its versions $Tmf_0(2)$, $TMF_0(2)$) denotes the spectrum obtained from the cover by the moduli stack of elliptic curves with level structure for the congruence subgroup $\Gamma_0(2)$ (the one that preserves the NS-R spin structure on an elliptic curve, see here).
Detailed discussion is in (Behrens 05).
This plays at least roughly the role for the Ochanine elliptic genus (the partition function of the type II superstring in the NS-R sector) as tmf does for the Witten genus (the partition function of the heterotic superstring). See at spin orientation of Ochanine elliptic cohomology for more.
After inversion of 6, $TMF_0(2)$ is (Behrens 05) the elliptic spectrum $Ell$ of (Landweber-Ravenel-Stong 93).
The periodicity of $TMF_0(2)$ is 8.
Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)
Mark Behrens, A modular description of the K(2)-local sphere at the prime 3 (arXiv:math/0507184)