Decapitation causes death through loss of blood to the brain. In humans, the period of anoxia following decapitation may be only a few seconds. Unlike humans, however, some species have a much higher tolerance for oxygen deprivation, allowing them to live much longer after decapitation, and rendering beheading an unsuitable method for euthanasia of these animals.
The best-known example of such an animal is the reptile. Cold-blooded, often amphibious, reptile brains are wired to withstand long periods without oxygen, and may retain consciousness for some time after decapitation. Clifford Warwick, author of Reptiles: Misunderstood, Mistreated, and Mass-Marketed (1990; Nower Productions, UK) wrote that reptile heads engage in conscious, non-reflexive activity for up to an hour after removal, "writhing in agony" from the "massive severance of tissue." They blink, attempt to bite, their eyes follow movement, and they even extend their tongues as if sampling the air for smells.
Humane methods for euthanizing reptiles, according to that author and others in the field, should involve either use of anaesthetic overdose or immediate destruction of brain tissue.
Cecil Adams, The Straight Dope columnist, was initially skeptical of claims that humans retained consciousness after decapitation, but a reader spun to him the following story of a car crash, in which his friend was decapitated:
My friend's head came to rest face up, and (from my angle) upside down. As I watched, his mouth opened and closed no less than two times. The facial expressions he displayed were first of shock or confusion, followed by terror or grief. I cannot exaggerate and say that he was looking all around, but he did display ocular movement in that his eyes moved from me to his body and back to me. He had direct eye contact with me when his eyes took on a hazy, absent expression... and he was dead.
Adams declared that he had re-opened his mind on the subject. (http://www.newtimesbpb.com/issues/1998-06-11/columns.html)