There is no one definition of a zombie as there are a few types. Some zombies possess a small intellect. The most common zombie used are those who shamble around and claw at things with only the desire to kill. Most have zero non-instinctual knowledge, other than what their hivemind or ruling overlord has for them.
Zombies may or may not be able to speak. Zombies may or may not be able to think. Do not be confused though. Zombies are created to be mindless. Zombies may or may not be able to run. Zombies may or may not possess motor skills such as opening and closing doors (would be for very advanced zombies). Zombies may or may not have rotting flesh and or missing limbs, but most commonly will. Depending on what type of zombie the person decides to play as they may or may not be able to bite, infect or otherwise turn another into a zombie. If the zombie can infect others, they cannot turn another undead into a zombie, as the zombie transformation is one of the weakest types and cannot over power other ones.