wow I feel for you I really do. AOL are pants, but anyway thats another story.
You didn't stay but I guess your Router has more than 1 network point to plug network cables into. i.e. more ports similar to what computer A is plugged into. The 2nd port on the router connect to computer B.
OK goto Start --> control panel --> network connections --> and locate your "local area network" right click this icon. is there an option to Enable?? if so select it. If there is no "Enable" then it already is.
click START --> run --> type "cmd" in the black windows type "ipconfig" you should get a response providing you with an IP address usually 192.168.0.2. Your default gateway or the Router will probably have an IP address of 192.168.0.1. as per the AOL helpdesk.
OK now type "ping 192.168.0.1" if you get a reply this is good news. now try and "ping
www.google.com" if you get a response you are on-line and have internet capability.
launch your prefered browser and surf to your heart contents.
If you have problems at this point go to (in IE) Tools --> Internet options --> connections --> LAN settings --> select "automatically detect settings" make sure all other options on that window are unchecked.
good luck