Equilibrium means the sort of condition that an undisturbed system will settle into after a long time. For example, gas molecules in a room in equilibrium will be spread about approximately uniformly, regardless of how uneven the initial distribution was.
As for the distribution of energies, it's certainly very possible (common, even) to be out of equilibrium. If say you have a pot of water that's hot on the bottom and cool on top, it's out of equilibrium. It doesn't have any one temperature.
What's perhaps more interesting is that even if some system is spatially uniform, it can be out of equilibrium. Take for example a system of nuclear spins in a magnetic field at low temperature. They tend to line up with the field. Now lower the magnetic field. The spins are in effect colder than they had been, since they're still lined up as much as they had been, but in a smaller field that alignment would, in equilibrium, only be found at a lower temperature. Meanwhile the atoms are still jiggling around at the original temperature. Even though the material is homogeneous, the distribution of what states it's in can't be described by any single temperature.
By the way, you may wonder what would happen in the example above. In effect the spins are colder than the atom motions. Energy is gradually traded between these modes until the system reaches a definite temperature, colder than the one it started at. This process is called cooling by adiabatic nuclear demagnetization, and it's a staple of the low-temperature experimental techniques. Before the new equilibrium is reached, the system simply doesn't have a temperature.
(published on 12/23/11)