The disease spreads through being bit by a carrier (which all zombies automatically are). Every alien species and even robots and machines can be infected with the disease and there's no cure.
At the end of the episode Sometimes You Can't Live with Dying, Baby, everyone in the entire universe becomes a zombie, resulting in a peaceful universe without sickness, death, or war.
Zombies are clinically dead, lacking any vital signs whatsoever. Despite this, zombies are still able to move and think. Seemingly the only way to truly "kill" a zombie is to shoot the infected victim through the brain.
Zombie skin (or fur) takes on a sickly greyish blue color. Their eyes are bloodshot, deeply sunk into their sockets, and have extremely contracted pupils. Zombies also lack a lot of control over their muscles, causing them to move sluggishly and be unable to fully close their mouths or speak, except through a complex language of groans which all sound the same to non-zombies.
New zombies have an intense craving for raw meat, which eventually goes away. They also have the inexplicable urge to congregate in malls.
These urges are references to zombie-based media, which often have flesh-eating zombies in malls, such as Dawn of the Dead, Left 4 Dead 2, and Dead Rising. However, these behaviors are very beneficial for the disease to manifest in its victims; a craving for raw meat and an urge to go to crowded public areas makes new zombies more likely to run into people and bite them out of hunger, thus ensuring that the disease continues to spread.