The old antivirus is missing things, I would guess. You don't have a firewall active that you know of? If you use Windows XP (These instructions may work for Vista) go to the Control Panel from the Start menu and find an icon that says "Security Center" or something similar. See what it says about firewall; if it only says the Windows firewall is active you need one.
I can recommend either ZoneAlarm's free firewall or Comodo's. I'd go with Comodo if you have no preference, for it includes some anti-malware properties:
ZoneAlarm (The "Get Basic PC Protection" side)
Comodo Personal Firewall
Make sure that you do not have any firewall (other than the Windows firewall) running before you install this! Having two active firewalls can be a big problem in terms of stability. When you install Comodo it will ask if you want to perform a scan for malware. Say yes and let if go; it may well detect and remove the nasties on its own. If so, you're good and can skip the next step if you want (although I recommend doing it).
Antivirus is more complex. Get a firewall installed first. Once that is installed download one of the following firewalls but DO NOT install it yet. These three are all good choices; pick one at random (Avira is a random pick from me if you're lazy):
avast!
AVG
Avira
Once the installer is downloaded, uninstall Norton. If you know the product you have (e.g. Norton Antivirus 2003) head to the
Norton Removal Tool page and download the appropriate uninstaller. If you just don't know what you have or it isn't listed there go to the Add/Remove Programs page in the Control Panel and remove it there.
Once Norton is gone install your chosen Antivirus program. Let it install and update itself (If it doesn't update itself look for a button that says "Check for update" or similar and run one yourself). If it asks to do a full system scan do it! Otherwise, open it and run a manual scan. I can tell you how if you are unable to decipher the control panel.
If these don't work we'll go to antimalware tools; hopefully this will be enough.