A zombie computer / zombie box is a computer that has been taken over.
Most, if not all, of your networking software (web browser, email client, IM client, or underlying software) has bugs that could allow malicious hackers to execute arbitrary code on your computer. These are exploited to do just that. Your computer will continue to operate as normal for you, but in the background, it's sending out spam (several years ago, it was estimated that 50-85% of all spam comes from zombies), it's attacking other computers (remember the 'Internet attacks' that took down Amazon, Yahoo!, and eBay? Or, more recently, Estonia.) and--of course--it's infecting other computers in order to expand its power. That's right: if you've taken anything from all those zombie movies you watched, it should be that zombies like to travel in hordes, and exploiting a security hole in popular software can lead to exponential growth and some truly massive botnets--some estimates put the largest at 50 000 000 compromised computers in 2007.
What can you do?
Nothing you haven't been told a hundred times, and probably ignored every time: be wary of email attachments, keep your software updated, use more secure and/or less popular products, and avoid unscrupulous websites (though the point of zombies is that they secretly take over scrupulous people's machines). Anti-virus and anti-malware software is of some use, but if I already control your computer from behind the scenes, I'll just replace your AV software with something that looks the same but doesn't actually detect me.
This time don't ignore the advice. Maybe you don't care so long as your computer continues to work for you, but zombie computers are attacking others like mad. If your computer steals my banking information for the Russian Mafia and you didn't take the minimal precautions to prevent that, I'm coming after you.
Sources and further reading