Firstly...
My O/S is Window's 5.1
I assume, we're talking Windows XP, then.
Secondly, turn your Windows firewall ON, for now.
Now, when you uninstalled Norton, it's very possible, it didn't remove everything, especially registry entries.
Go Start>Run, type in:
regeditHit Enter.
Registry Editor will open.
Go File>Export, and save your registry file to known location.
Go Edit>Find
In "Find" field type in:
NortonClick OK.
When entry is found, right click on it, click
DeleteHit F3 to find next
Norton related entry. Right click, Delete.
Continue, until no more
Norton entries are found.
Repeat same action, looking for:
SymantecNow, you need a firewall. I'm not a big fan of ZoneAlarm, so I'll recommend Comodo:
http://www.personalfirewall.comodo.com/Download it.
Turn your Windows firewall OFF.
Install Comodo.
Restart computer to see, if everything checks out.