What is the Hubbard-Holstein model? Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for.
 A: The Hubbard-Holstein model is a electron-phonon model with Hamiltonian
$$ H ~=~ -t\sum_{i{\delta}\sigma}c^{\dagger}_{i\sigma}c^{}_{i+\delta\sigma}
        +U\sum_{i}n_{i\uparrow}n_{i\downarrow}  \nonumber +\omega_{0}\sum_{i}b^{\dagger}_{i}b^{}_{i}+g\sum_{i\sigma}n^{}_{i\sigma}(b^{\dagger}_{i}+b^{}_{i}).$$
Quoting R. Ramakumar, A. N. Das, Polaron cross-overs and d-wave superconductivity in Hubbard-Holstein model, arXiv:cond-mat/0611355,

Here $t$ ($>\,0$) is the hopping energy between molecules at lattice site
  $ {i}\,$ and its nearest-neighbor lattice sites $i\,+\delta$,
  $c_{i\sigma}$ ($c_{i\sigma}^{\dagger}$)
  is the annihilation (creation) operator for the electron with spin
  $\sigma$ at a lattice site $i$ and $n^{}_{i \sigma}$ is the corresponding number
  operator, $U$ is on-site Coulomb repulsion,
  $b^{}_i$ ($b_{i}^{\dagger}$) is the phonon annihilation 
  (creation) operator, 
  and $g$ is the interaction strength.

The Hubbard-Holstein model a hybrid between the Hubbard model (which has no electron-phonon coupling $g=0$), and the Holstein model (which has no Coulomb repulsion $U=0$).
See also p. 60 in the 2003 thesis  Phonons, charge and spin in correlated systems by Alexandru Macridin. The pdf file is available here.
