A zombie is a machine that has been added to a botnet. A botnet is a group of machines controlled by criminals. Generally, these computers are used to send spam. The botnet can also be rented to other criminals for use in denial-of-service attacks.
Your computer must be infected with malware to be turned into a zombie. The software runs surreptitiously. That's because the criminals want to use your machine as long as possible. They don't want you to know.
Given this, it is often difficult to tell if your computer is a zombie. Your computer may be sluggish. There could be a lot of causes for this, though. You would probably also notice new programs on the machine.
Access to security sites could be blocked. And your connection speed could slow to a crawl. Most people don't realize their computer is infected.