A zombie is a reanimated dead body. They mainly appear in Dead Beat.
They aren't a real person with thoughts and feelings, the corpse is used to being a person: to eating, breathing, and having a beating heart. The necromancer controls them by recreating the heart beat by playing a beat or some kind of rhythmic music.
Zombies are grey-faced and show generally a purpose in their movements. They are used as foot soldiers by the necromancer controlling them. Human bodies are used rather than animal bodies because humans leave a stronger "metaphysical impression". A necromancer can pour more energy into it—"they make stronger goons".
Older corpses leave a deeper metaphysical impression. They are harder to call up but are stronger, easier to control and they don't damage as easily—"stronger undead flunkies". 
Although necromancy is forbidden by the Fifth Law of Magic, the Law only forbids wizards reanimating people, not necessarily animals. The reason being that Necromancers only use human bodies while sane Wizards don't touch the practice at all.
Major trauma should kill them eventually, even though a gun or a bat won't do much against them.  They can be burned to ashes, chopped up with an ax, run over with a big truck. They can also be stopped by cutting off the flow of energy that animates them.; running water (at least the size of a small stream), or a magic circle can do that. Removal or destruction of the drum that animates them is also effective.
In the seriesEdit
In Dead Beat, Grevane attempted to kidnap Waldo Butters at the Forensic Institute with four zombies—one of them the newly created Phil, the security guard from the front entrance. Harry explains to Butters what Zombies are and how to stop or kill them. Grevaine had caddy outside with a giant bass woofer and he beat a book against his leg while in the lab, which is how he controlled and commanded them.
Grevane led a charge of Zombies that attacked Harry Dresden's apartment. They badly damaged the steal security door but never broke it down. Waldo Butters was stolen by Zombies out the side sunken window. Corpsetaker charged Grevane's army of Zombies with her own army of Specters. Harry answers more Zombie questions from Butters.
Grevane entered the Field Museum of Natural History with his marching army of Zombies and took the Word of Kemmler from Harry leaving him with Quintus Cassius. Harry reanimates Sue into a Zombie T-rex to stop the necromancers' Darkhallow. Waldo Butters provides the beat with his polka drum suit. Captain Luccio, Carlos Ramirez, Yoshimo, and Kowalski were protecting a group of children against a mass of Zombies.