Most people use the term 'virus' to also refer to computer 'worms', which are executable programs that copy themselves from computer to computer using security holes. Most 'viruses' these days are just that, a worm.
'Worm' sounds a bit more innocent than virus, but the reference is to parasitic worms that live inside living things, not the ones living in the ground. Computer worms are close to them in that there will only be one copy of them, they are much larger, and they don't make other programs do they copying for them, but are full programs. These come in via open ports, which a firewall can help with, or malicious web pages or email messages, from other machines that are infected.
Worms, on the other hand, hardly ever do anything noticable to your computer except slow it down and cause crashes. People often have a computer worm without knowing it, and that is by design. The first thing the worm does is start attacking other machines to look for the same security hole on that machine that they came in on. Then they turn your machine into what is called an '0wned' (meaning 'owned', with a zero at the start) machine, a machine controlled by hackers. The hackers can read every bit of data on it, and they lease it to spammers, causing you to spam others. (In addition to any worm-laden email you might be already sending.)
Why the change? Simple. Money. Viruses used to be written for fun, by bored programmers. Often they'd be contests to see how small they could be, or how tricky they could avoid detection by antivirus. There were actual toolkits to produce viruses that would guide you through the process.
Enter the internet. Suddenly, worms that spread themselves automatically were possible. The first one was a proof of concept more than anything else.
Then, enter spam. Suddenly, thanks to spam-fighters blocking any computer spammer set up, having a million machines became extremely valuable. So spammers, who were never that ethical to start with, started hiring hackers to create worms that take control of other machines, creating 'Owned' machines.
And the threat was off and running, and, yet, harder than ever to see. Almost all publicity is still on the 'created-for-fun' worms that actually do something noticable to your computer, while dozens of silent intruders sit on unwitting user's machines, relay spam, attack other machines, transmit personal information back to their controller, and silently control tens of millions of infected machines.

