Has someone hacked into my computer?

I began getting notices from my Norton intruder protection system that a port-scan had been blocked. I checked the history and found that my computer had been attacked over 10 times all from different IP addresses by someone trying to run a port-scan. This has happened a couple times since. Since the attacks... show more I began getting notices from my Norton intruder protection system that a port-scan had been blocked. I checked the history and found that my computer had been attacked over 10 times all from different IP addresses by someone trying to run a port-scan. This has happened a couple times since. Since the attacks keep coming over and over, I've started unplugging my internet connection as soon as I start seeing warnings.

Yesterday, my daughter was watching a movie online when the internet shut down. I tried to get the internet running again, but even though my computer said it was connected to the internet, I couldn't get it to work. Finally I discovered it was my Norton firewall that was blocking the connection. When I turned it off and turned on the windows firewall, my internet started working again.

I'm very worried about using my internet under these conditions, since I know someone is trying to attack my computer. I'm also wondering why my Norton firewall suddenly made it impossible for me to use the internet since I have been using it all along. I'm wondering if this was engineered by whoever is trying to get into my computer. If anyone has any information on what might have happened, how I can find out if someone has access to my computer, and how to fix my Norton program, I would appreciate the help.
Update: Thanks for the great answers so far. I have one concern about changing antivirus programs, though, and that is they are notorious for slowing down computers. I remember the one I installed before slowed my computer down so much I had to wait 5 to10 minutes to use it after turning it on. Even after I uninstalled... show more Thanks for the great answers so far. I have one concern about changing antivirus programs, though, and that is they are notorious for slowing down computers. I remember the one I installed before slowed my computer down so much I had to wait 5 to10 minutes to use it after turning it on. Even after I uninstalled the program, my computer stayed slow. I have a new computer now and don't want to slow it down. Any additional suggestions?
Update 2: Thanks, Chuck. I will have to email those instructions to my brother and have him walk me through it because that's all greek to me. It sounds like it will help though.

Any additional info on firewalls that don't greatly slow down the computer but still work well would still be helpful.
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