Start by deleting all your network connections in the control panel. Then run the network wizard specifiying connected through a LAN (if its XP I think the option is the 2nd bubble, connected through a residential gateway)
Make sure TCP/IP is setup to automatically receive an IP address and make sure the router has the DHCP server enabled. Make sure u restart changing the settings too
Also make sure its not a hardware issue... try the cable from the working computer to eliminate that as a possibility. NIC drivers are a possible problem as well but rather unlikely
As for wireless.. disable IEEE 802.11x authentication, at least until you get the basic connection working. Also make sure Ad-Hoc mode is not enabled