As seen in ...
Resident Evil IV
What are they?
Parasites that turn victims into mindless, zombie-like slaves are fairly common in nature. There's one called toxoplasmosa gondii that seems to devote its entire existence to being terrifying.
This bug infects rats, but can only breed inside the intestines of a cat. The parasite knows it needs to get the rat inside the cat (yes, we realize this sounds like the beginning of the most fucked-up Dr. Seuss poem ever) so the parasite takes over the rat's freaking brain, and intentionally makes it scurry toward where the cats hang out. The rat is being programmed to get itself eaten, and it doesn't even know.
Of course, those are just rats, right?
How it can result in zombies:
Hey, did we mention that half the human population on Earth is infected with toxoplasmosa, and don't know it? Hey, maybe you're one of them. Flip a coin.
Oh, also, they've done studies and shown that the infected see a change in their personality and have a higher chance of going batshit insane.
Chances this could cause a zombie apocalypse:
Humans and rats aren't all that different; thats why they use them to test our drugs. All it takes is a more evolved version of toxoplasmosa, one that could to do us what it does to the rats. So, imagine if half the world suddenly had no instinct for self-preservation or rational thought. Even less than they do now, we mean.
If you're comforting yourself with the thought that it may take forever for such a parasite to evolve, you're forgetting about all the biological weapons programs around the world, intentionally weaponizing such bugs. You've got to wonder if the lab workers don't carry out their work under the unwitting command of the toxoplasmosa gondii already in their brains. If you don't want to sleep at night, that is.
You may be protesting that technically these people have never been dead and thus don't fit the dictionary definition of "zombies," but we can assure you that the distinction won't matter a whole lot once these groaning hordes are clawing their way through your windows.