Different stories have different rules. In Return of the Living Dead, a severed zombie arm would continue to try and attack the living. Sometimes zombies run. Sometimes they can't, and explanations for the outbreak are different every time. That's part of what makes them so frightening, but the defining characteristics of zombies are pretty consistent and fairly straightforward:

1. Zombies are reanimated corpses that don't think or feel. All they want is to eat the living.

2. Destroy the brain, kill the zombie.

3. If a zombie bites you, you die and become one of them.