How long does a human head remain conscious after being decapitated?
The current medical consensus is that life does survive, for a period of roughly thirteen seconds, varying slightly depending on the victim's build, health and the immediate circumstances of the decapitation. The simple act of removing a head from a body is not what kills the brain, rather, it is the lack of oxygen and other important chemicals provided in the bloodstream. To quote Dr. Ron Wright "The 13 seconds is the amount of high energy phosphates that the cytochromes in the brain have to keep going without new oxygen and glucose" (Cited from urbanlegends.com, no longer extant). The precise post-execution lifespan will depend on how much oxygen, and other chemicals, were in the brain at the point of decapitation; however, eyes could certainly move and blink.
Do You Remain Aware?:
This solely technical survival forms only part of the answer; the second question is 'how long does the victim remain aware?' While the brain remains chemically alive, consciousness can cease immediately, caused by the loss of blood pressure or if the victim is knocked unconscious by the force of the decapitating blow. If that weren't to happen immediately, an individual could in theory remain self-aware for part of the thirteen-second period. There is no consistency in this answer, as the precise length of both actual, and practical, survival will vary depending on the victim. Of course, this applies to many forms of swift decapitation, and not just to the victims of the guillotine.