The Bose-Hubbard is a lattice model. It's quite a rich model, and can only be solved exactly in certain limits. Apart from being a relatively simple model to write down, it turns out to contain a wealth of funky physics, and people apply all sorts of techniques to understand the behavior of the model in certain limiting cases.
The model has a number of applications: it turns out to approximate the behavior of electrons in a lattice, and it is also related to superconductivity.
The basic rules of the game are as follows:
(1) consider a lattice -- it can be defined on any lattice with an arbitrary dimensions, but for simplicity just imagaine the two-dimensional square lattice.
(2) Each lattice point is either occupied by a boson or not. (The Hubbard model is the case where you would talk about fermions instead of bosons). The total number of bosons present is not fixed, and can take on any value. But only for particular values (i.e. half filling: half as many bosons as there are lattice points) are we able to perfroms some of the calculations (the number of bosons can be fixed by inserting a chemical potential term -- this is a standard technique in condensed matter physics)
(3) The Hamiltonian of the system has two "terms" (each term is a summation). A hopping term and an interaction term.
(4) The hopping term is like a kinetic energy term. A boson can hop from one lattice point to the next, and this term measures the energy associated to that process. The difference with a normal kinetic term is that it actually favors hopping of the bosons: the more the bosons hop around, the lower the energy of the system is. The strength of the hopping is given by t.
(5) The interaction term, on the other hand, is an on-site interaction. It only contributes to the Hamiltonian if there are two bosons on the same site. In the interesting case this interaction is taken to be repulsive: if two bosons occupy the same site, this gives a positive energy contribution. To lower the energy you would want a maximum of one boson per site. The strength of the repulsion is given by U>0
This is basically the model. Now you can start to tweak the parameters. As you can see, the hopping and interaction term are competing against each other. If the hopping term is dominating (t>U) the bosons roam around freely. If the interaction term is dominating, the behavior of the bosons highly depends on the number of bosons. If there are an equal number of bosons as there are lattice points, then the ground state of the system equals the case where there is exactly one boson per site. If you are at half-filling things start to get a little more interesting.
