Why do octopus tentacles still move after they are cut off?

The octopus nervous system is different from a mammal's. The brain of the octopus does not have complete control over the body. The tentacles of the octopus contain ganglia which can operate separately from the octopus body. The octopus does not directly control it, but instead sends commands to the arms and lets them do the work. Octopus have little to no proprioception.

When an octopus loses an arm, it can no longer control it (obviously). Because the ganglia are still intact, it is still capable of operating independently of the octopus. The independent arms can react to stimuli as it did when it was still connected to the octopus, though it has not been proven that the arms themselves are actually self-aware and not just automatic responses.

If lost, an octopus can regenerate an arm. This occurs during mating, when an octopus's Hectocotylus (mating arm) often breaks off and is left in the female.