However, despite the myth, you are not conscious. The tradition is to call the beheaded's name after they lost their head. However, scientists in the 1700s noticed that the heads would respond by blinking and opening their eyes! However, the effect only lasts for 2-3 seconds. On July 17, 1793, Charlotte Corday was decapitated for murdering popular journalist Jean-Paul Marat.
The interesting part about this is that after her head fell, one of the executioners picked up the head and slapped her cheek. Corday's eyes darted to the executioner's face in anger. Some report that blinking after death can occur for up to 30 seconds after beheading!
However, this is not your brain still being conscious. Brain scientists believe that there is still enough brain activity for involuntary reflexes to create this kind of phenomenons. To read more on decapitation and more brain myths explained, check the source