It sounds like your on a high speed internet connection and you have your computer either directly connected to the modem itself, a hub, or a switch.
With any high speed internet connection nowadays you pretty much have to have a router with a built in firewall.
You can get a D-Link DI604 or 704P pretty cheap.
This problem is most common on newer computers as the new viruses are designed to take advantage of current exploits, and Norton although it detects and removes viruses can not prevent them from coming in, as a lot of these new viruses don't rely on you to open files or run programs, they just scour the range of public IP addresses on the net, and check each IP in those ranges for open ports and security holes.
The reason a Router solves this problem, is because instead of your PC getting a publich range IP address from your ISP, the router gets that IP address. Your PC in turn gets a private range IP address directly from the router.
This way nasty viruses and such can only talk to the router and not your PC.