The Fermi surface is simply the surface in momentum space where, in the limit of zero interactions, all fermion states with (crystal) momentum $|k|<|k_F|$ are occupied, and all higher momentum states are empty. Amazingly, Luttinger and Ward showed that the Fermi surface survives even with interactions to all orders in perturbation theory (Oshikawa later showed this nonperturbatively, see also the arXiv version).
The point of the Fermi surface is that this is where all of the low-lying excitations of the system live -- the Fermi energy is so much larger than room temperature that for room-temperature experiments, all of the thermodynamics is dominated by excitations right at the Fermi surface, and thus knowing its structure is very important.
A more advanced reason for the importance of the Fermi surface is that by knowing its structure (look up "Fermi surface nesting"), we can understand the instabilities of a metal, for example, for sufficiently low temperature, a normal metal settling into a charge density wave state.
This post imported from StackExchange Physics at 2014-04-01 17:30 (UCT), posted by SE-user wsc