You own your computer, of course. You bought it. You paid for it. But how much control do you really have over what happens on your machine? Technically you might have bought the hardware and software, but you have less control over what it's doing behind the scenes.
Using the hacker sense of the term, your computer is "owned" by other people.
It used to be that only malicious hackers were trying to own your computers. Whether through worms, viruses, Trojans or other means, they would try to install some kind of remote-control program onto your system. Then they'd use your computers to sniff passwords, make fraudulent bank transactions, send spam, initiate phishing attacks and so on. Estimates are that somewhere between hundreds of thousands and millions of computers are members of remotely controlled "bot" networks. Owned.