What is Fatigue?

In short:

A state of serious tiredness and exhaustion.

Defined by ICAO (Annex 6, Part I, as amended in 2009):

"A physiological state of reduced mental or physical performance capability resulting from sleep loss or extended wakefulness and/or physical activity that can impair a crew member's alertness and ability to safely operate an aircraft or perform safety related duties."

Explained by medical experts:

A state, not related to an illness, resulting in a decreased ability to maintain function or workload due to mental or physical stress. It has also been described as the progressive decline in alertness and in the ability to process information. There are two major factors inducing fatigue: sleep loss and circadian rhythm disruption (jet-lag and irregular nigh/day work).

See "Pilot Fatigue", Dr. Samuel Strauss