Chickens can run after their heads are cut off because often a chicken's head is cut off at the base of the skull, leaving the brain stem intact, which controls the nervous system and reflex actions. This enables the chicken to move around for a while, but usually the bird stops moving within minutes of being beheaded.
Although a chicken doesn't usually last long after its head is cut off, Reference.com points out that Miracle Mike was the exception to this rule and lived for more than a year after parting ways with his head. Usually an ax that beheads a chicken also severs its jugular vein and the chicken soon bleeds out. The ax that took Miracle Mike's head, however, nearly missed the jugular vein. This, along with a lucky blood clot, prevented him from bleeding out.
For 18 months after September in 1945, Lloyd Olsen kept Miracle Mike alive by feeding and watering him with an eyedropper straight into his esophagus. Not only did this method of feeding keep Miracle Mike alive, but he also thrived, nearly quadrupling in weight over that time. Olsen then took "The Headless Wonder Chicken" on the road as a sideshow where people paid to see him. He died after choking on something.