Fermi Level is the energy level at which an average of 50% of the available quantum states are filled by an electron. It was named after the famous physicist Enrico Fermi, who had a significant hand in developing the atom bomb.

The Fermi Level relates the probable location of electrons in a band diagram. If you are looking at a band diagram of a substance (usually a semiconductor) the Fermi Level tells you where the average electron is. For metals the Fermi Level lies in the conduction band and for insulators the Fermi Level lies in the valence band and for semiconductors the Fermi Level lies in the band gap.

Semiconductors are unique because the Fermi Level lies in the band gap which cannot contain electrons. This, however, doesn't prevent the statistical location of the Fermi Level lying in the band gap.

|    Conduction Band   
| -----------------------
|       Band 
|   - - - - - - - - Fermi Level
|        Gap
| -----------------------
| O O O O O O O O O O O O 
|     Valence Band \
|                   \
|                  electrons
+------------------> x direction
Note how the valence band is full of electrons and there are relatively few electrons in the conduction band, placing the Fermi Level right in the band gap.

It is worth mentioning that when a piece of semiconductor (or any substance) is at equilibrium, with no net current or applied field, then the Fermi Level will be flat.

When you create the band diagram for various semiconductor devices, like a p-n junction, the band diagram will have slopes and curves in it, but the Fermi level remain flat. A sure fire way to tell if there is an applied voltage is to look for a sloping of the Fermi Level.

The Fermi Level is important also, for determining the type of a semiconductor. If it is p-type then the Fermi Level will lie closer to the valence band. If it is n-type then it will correspondingly be closer to the conduction band. It is very useful because it will tell you the most probably location of an electron, which is useful information if your trying to govern their flow.