One truly remarkable feature of statistical physics (and condensed matter physics in particular) is the emergence of phase transitions. When dealing with large numbers of particles one often finds that, as a function of some parameter like temperature or pressure, the whole collection of particles can undergo a change of state. For example, as liquid water is warmed through 100 C at atmospheric pressure, it boils into a vapor phase of much lower density, even though it is still made up of the same water molecules as before. Understanding how and why phase transitions take place has kept many physicists occupied for a long time.
Of particular interest is understanding how microscopic interactions (e.g., polar attraction between individual water molecules) connect to the phase behavior. A classic toy model of this is used to examine magnetism. It's a comparatively simple statistical physics problem to understand how a single magnetic spin (in real life, something like one of the d electrons in iron) interacts with an external magnetic field. The energy of a magnetic moment is lowered if the magnetic moment aligns with a magnetic field - this is why it's energetically favorable for a compass needle to point north. So, one does the statistical physics problem of a single spin in a magnetic field, and there's a competition between this alignment energy on the one hand, and thermal fluctuations on the other. At large enough fields and low enough temperatures, the spin is highly likely to align with the field. Now, in a ferromagnet (think for now about a magnetic insulator, where the electrons aren't free to move around), there is some temperature, the Curie temperature, below which the spins spontaneously decide to align with each other, even without an external field. Going from the nonmagnetic to the aligned (ferromagnetic) state is a phase transition. A toy model for this is to go back to the single spin treatment, and instead of thinking about the spin interacting with an externally applied magnetic field, say that the spin is interacting with an average (or "mean") magnetic field that is generated by its neighbors. This is an example of a "mean field theory", and may be solved self-consistently to find out, in this model, the Curie temperature and how the magnetization behaves near there.
Mean field theories are nice, but it is comparatively rare that real systems are well described in detail by mean field treatments. For example, in the magnetism example the magnetization (spontaneous alignment of the spins, in appropriate units) goes like (1-T/TC)1/2 at temperatures just below TC. This is not the case for real ferromagnets - the exponent is different. Because of the nature of the approximations made in mean field theory, it is expected to be best in higher dimensionality (that is, when there are lots of neighbors!). Here's a question for experts: what real phase transitions are well described by mean field theory? I can only think of two examples: superconductivity (where the superconducting gap scales like (1-T/TC)1/2 near the transition, just as mean field theory predicts) and a transition between liquid crystal phases. Any others?