Everybody loves zombies. You can shoot your math teacher as soon as he turns into an undead walking creature with the excuse it's an enemy for the living and that you're killing an already-dead corpse. You can shoot some 10 of them and safely run away to reload your rifle. You can kick them and see their buttocks fly away.
Anatomy of a Zombie
A Zombie core clichè may be like this:
Zombie (telling living from dead, wanting to eat you, moving slowly)
This is, as said, just a core. Depending on the setting and on the way you think of zombies, this may vary. This average zombie can still be used as-is in in non-zombie-specific settings. Their clichè level can have the usual range 1-6 depending on how dangerous they are from a Newly Born Zombie (1) to a Zombie Lord (6).
Basically a Zombie is a very stupid walking creature wanting to eat living beings. As appetizer they usually have brain. They appear to know by instinct who is a zombie and who should be. They're not fast, but they tend to move in groups of too-many-for-my-ammo. Occasionally, they have rudimental weapons like axes or choppers. Chainsaws are often found in zombie settings, to split them apart or even carried by them. For some strange reason, they appear to be somehow related with leprous and a have a propensity for losing pieces of flesh and limbs. They also have a terrible taste for clothing and seem to take inspiration from post-transformation Hulk (the incredible one).
I also guess they stink of rotting flesh, but I'm not sure of it.
If one gets bitten by a Zombie, but does not become its dinner, chances are she will turn into a zombie as well. The best way to prevent your best friend to become a zombie once bitten is to carefully separate his head from his neck. Exception for this are treated in the "Origin Of Zombies" section. The time from "Gee, he got me!" to "UUUUUUoooooohhhh" is up to the GM. Generally they turn into a Zombie in the less appropriate moment (when the group is stuck into an elevator, for example). Or within half an hour or a couple. Whatever.
Types of Zombies
There are various kinds of Zombies down there, these are the most common. Every Zombie clichè, as said, can appear in different formats (1-6), but remember the rule of thumb: the more they are, the less they're strong (unless you belong to the Evil Game Master specie or one of the players stole your girlfriend). So, let's welcome hordes of Stinky Michael Jackson Looking Zombie (1), but be careful of using a thousand You Read Into My Eyes I'm Gonna Eat You Zombie (5).
This is the average zombie. He walks saying "Uuuuuuooooohhhh" and is attracted by the tasty flesh you happen to carry with you. A good shot in his forehead would make it a Dead UnDead (0). As a special feature, he suddenly remembers he can actually move faster when a player is near (say some one or two meters) and tries to catch her (treat this like a pump).
Carl Lewis Zombie
This is a zombie seen, for example, in 28 Days Later(c) by Danny Boyle. Not only they never forget they can move fast, but they have pieces of jaguar DNA into them.
A former-pronstitute, a former-filling station attendant, a former Osama Bin Laden. When you look at this zombie it is soon clear that he had a life before having a death and that he had a work. Indeed, this is just like the Zombie-Zombie, but better shaped-out and with some twists. A Former Lumberman Zombie (3) would try to kill you with his axe, a Former Geography Teacher Zombie would try to tell you what the main cities of Sudan are talking directly into your brain.
This is usually a Zombie Leader (6). A Good example is the one in Romero's Land of The Dead(c). Leader of a group of Zombies, he's stupid but not *that* stupid. Say he's smart enough to understand that a glass wall can be broken to take the people hiding behind it or to appreciate a soap opera. Because of this his followers can perform a little bit more intelligent actions. Whenever the Zombie Lord dies, the others go back to their no-brained existences.
Aunt Mary Zombie
This is a must have. Remember to include, here and there, a zombie-ified relative.
According to some scientist, bitten animals can turn into zombies as well. Remember the doberman in Resident Evil. No use to call them Fuffy or other stupid names, if you try to pet them you'll become a One-Handed Stupid Guy (2). Throwing sausages to distract them is also useless, you'd better throw your left foot. Any beast can do the work, from dogs to cats, rats, bats. I'd love to see a zombie chimpanzee.
This zombie is not really a nerd, but the GM almost certainly is. He's running a Zombie game but he's in need of trolls, goblins and beholders. So, here come monster zombies. Imagine them like you prefer, but they gotta be huge, ugly and nasty. Possibly Five Eyed John Cena Sized Dribbling Roaring Manticore-Like Zombie (6). Big amounts of them in every Resident Evil related product.
This is a classic. He's just gained 5 bullets from your magnum and he's lying down on the floor. You approach the corpse to loot it just to discover he's not a Dead UnDead (0) but an AlmostDead UnDead (1), with just that little life into him to allow him to bite you one last time. Any Zombie can be an AlmostDead UnDead before definitely die.
Origin of Zombie
But... Where do they come from? Hypothesis on the subject are numerous.
They have been magically created by some sort of evil wizard with an attitude toward necromancy. If you want to end the game in a joyful way, invent something stupidly hippy like once the wizard is killed, all the zombies and de-zombiefied. They're usually created via an enchantment, a wand, a spell. Usual stuff, but the sound of the sorcery is always SWOOOFFFFF.
Mad Scientist Experiment Zombies
They're the brainchild of a mad scientist, an evil experiment, a para-governmental occult organization called Parasol or similar. This is a more "scientific" approach to zombies. Think about making them out of radiations, viruses and things like that. This also means there's probably a hard-to-find cure to the disease. Nerd Zombies are also common, if you choose this option.
Who Gives a Damn, Shot & Run! Zombies
Needless to say, the most common ones.
Running a Zombie Campaign
Zombies are good if thrown in any fantasy or horror settings among vampires and werewolves. If you're not going for something original, of course.
But they're also great for a Zombie-centric game. Modern world is perfect for the purpose. Choose a large city, so there are more people to turn into zombies and more zombies to kill.
Running a Zombie Campaign is easy. The characters wake up and find out that there exists no more a civil (?) society but everyone apart from them is walking strangely with a queer appearance and mostly is trying to make a supper out of them.
The characters apparently have never turned on the tv in their lives, never been to the cinema, never roleplayed, never bought comic books so they don't instantly realize there's a zombie outbreak going on and it takes a while to understand it.
Characters' main aim is (or should be) to save their butt. Optionally find a cure to save the world or just escape in Bangladesh or a far, far place to start over a new zombie-less life.
The main plot elements are the PCs running and shooting. This may be boring or funny according to the place where they run and shoot and the kind of zombies. This is up to the GM. Vary the locations, put them in trouble.
There are essentially two ways of doing the things: loads of zombies / loads of weapons or few zombies / few weapons. Loads of zombies / few weapons is an option too.
In the first kind of game the PCs find often a weaponry, a police-station and similar places to take kilos of ammo they'd eventually move into zombie corpses.
In the second kind of game, there are fewer zombies, but they're angrier and above all there are very few weapons and ammo. It's hard to find them and PCs (gamers, actually) swear every time "one shot, one kill" does not work. Think at any Resident Evil game (any but Resident Evil 4).
If you're obsessed with background and narration, there must be a way to stop the plague. Invent a decent arci-enemy controlling the whole thing and having the key (magic medallion, sonic weapon or chemical antidote) to stop it. Throw in some non-zombie foes like white collars supporting The Enemy, bribed military police and so.
That's it. And remember: have fun, shoot and run.. Fast!