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The Hubbard model is the simplest model of interacting electrons on a lattice: H = -t sum_{<ij>,sigma} c^dagger_{i,sigma} c_{j,sigma} + U sum_i n_{i,up} n_{i,down}. The first term is nearest-neighbor hopping (kinetic energy, bandwidth W ~ zt), and the second penalizes double occupancy of any site by Coulomb repulsion U. The competition between kinetic energy (which delocalizes electrons) and interaction energy (which localizes them) produces a rich phase diagram including metallic, Mott insulating, antiferromagnetic, and (in some geometries) superconducting phases. At half-filling with U >> t, the model reduces to the Heisenberg antiferromagnet. The Hubbard model is believed to capture the essential physics of high-temperature superconductivity in the cuprates.
The Hubbard model is to strongly correlated electron physics what the Ising model is to statistical mechanics: the simplest possible model that captures the essential competition. It was introduced independently by Hubbard, Gutzwiller, and Kanamori in 1963, and it contains just two parameters. The hopping integral t measures the amplitude for an electron to tunnel between neighboring sites (kinetic energy, favoring delocalization). The on-site repulsion U penalizes having two electrons (with opposite spins) on the same site (interaction energy, favoring localization). The tension between these two tendencies produces the rich physics of correlated electrons.
For U = 0, the Hubbard model is just the tight-binding model — non-interacting electrons forming energy bands. Band theory says a half-filled band is metallic. For U >> t at half-filling, every site is singly occupied and charge fluctuations are frozen out — the system is a Mott insulator with a charge gap of order U. This is the fundamental failure mode of band theory: interactions can make an insulator out of what band theory predicts is a metal. Transition metal oxides like NiO, CoO, and V_2O_3 are Mott insulators, and their insulating behavior puzzled physicists until Mott's insight that Coulomb correlations are responsible.
In the Mott insulating limit (U >> t, half-filling), the remaining degree of freedom is the spin on each site. Virtual hopping processes (electron hops to a neighbor and back, through a high-energy doubly-occupied intermediate state) generate an effective antiferromagnetic exchange J = 4t2/U between neighboring spins. The half-filled Hubbard model at large U thus maps onto the Heisenberg antiferromagnet, explaining why Mott insulators are so often antiferromagnetically ordered. This connection between charge localization and magnetic ordering is one of the central insights of correlated electron physics.
The most exciting and unsolved regime is the doped Mott insulator: start from the half-filled Mott state and remove some electrons (or add some holes). The doped holes can move through the antiferromagnetic background, disrupting the magnetic order. In the 2D Hubbard model on a square lattice, there is strong numerical and analytical evidence that the doped system develops d-wave superconductivity — the same symmetry observed in cuprate high-T_c superconductors (YBa_2Cu_3O_7, La_{2-x}Sr_xCuO_4, etc.). Whether the Hubbard model rigorously supports superconductivity in 2D, and if so with what T_c, remains one of the great open questions. The model also exhibits stripe phases (interleaved charge and spin order), pseudogap behavior, and other phenomena seen in cuprates. Solving the 2D Hubbard model is simultaneously one of the most important and most difficult problems in theoretical physics.