Great question! There are several ways for a virus to spread from one computer to another. Let's look at those, and ways to prevent the propagation of pestiferous programs.
Careful With That Paperclip...
The most common is via an email attachment. I'm not aware of any current exposures in popular email software that allow a virus to spread simply by opening or reading an email.
But plenty of people are still willing to open anything that pops into their inbox. Opening an attachment without verifying its contents with the sender is asking for trouble. Never run a program that you save to disk from an email attachment without anti-virus protection in place. The same goes for Word documents, which can have viruses in embedded macros.
Dubious Downloads
Another common way for a virus or spyware to spread is by piggybacking on other software that you download. If you just can't resist the latest toolbar, file-sharing gizmo, coupon dispenser or email enhancer, you may be at risk. Often these and other downloads come with malware, free of charge. Running a good anti-spyware program will reduce your risk.
Just Click NO!
Ever been to a website and gotten one of those "Click here to install and run {Whatever}" popups? Just click NO! Unless you know for sure that you need to press OK (such as when downloading security updates from Microsoft) then treat these popups as threats to your computer.
Sometimes It's NOT Nice to Share
If you are on a network, and you allow your hard drive to be shared (enabling other network users to read and write your files) then you are WIDE OPEN to the stupidity of the other users on the network. If any one of them got a virus, there's nothing to prevent it from copying itself to your computer via the network share. Unless you absolutely have to share your hard disk, don't. You can check to see if you have enabled sharing by right clicking on the drive icon in My Computer, then select Sharing.
The SneakerNet Menace
And don't forget portable media, such as floppies, CDROMs, and those nifty little flash drives that connect to a USB port. Assume that any file you copy from external media could be infected. Keeping your anti-virus protection enabled and updated should protect you here.