In any system that obeys Fermi-Dirac statistics (most usually electrons in a conductor or semiconductor) it is the energy of the most energetic particles in that system at absolute zero.
In Fermi-Dirac systems, notably electrons in a semiconductor, the electrons are pictured as progressively "filling up" the available energy levels prescribed by quantum mechanics, until all the electrons are accounted for. At absolute zero, the electrons would therefore completely fill the levels up till some critical energy at which there are no more electrons left. This results in a distribution of energies which goes up as the square root of energy and then suddenly drops to zero at the Fermi energy
At non-zero temperatures, the distribution starts off the same at low energies, and then rather quickly plunges (but not absolutely vertically) towards zero around the Fermi energy. If I recall correctly, the Fermi energy is at or near the half-way point down the quick plunge.
The Fermi level is extremely important in explaining and designing semiconductor materials. In an intrinsic semiconductor the Fermi energy is about half way between the top of a low conduction band and the bottom of the next higher band. It is then pushed either up or down by doping the semiconductor with appropriate materials to create p-type or n-type semiconductors.