On the computer that refuses to connect correctly to the Internet via the wired connection to the router, right-click on the little connectivity icon down by the clock on your Windows taskbar (if you have WinXP, WinVista, or Win7).
Choose to open the Network & Sharing applet.
Remove your current network
Set up a "new" connection to the network, entering the WPA2 (or whichever security scheme your network uses) password in the box that prompts you for that information.
Note that Windows XP, and Windows Vista may not use the term "Network and Sharing"; my answer is based on what I see in Windows 7. But since you'll only have a couple of choices at most when you right-click on the connectivity icon down by the clock on your Windows taskbar, it shouldn't be too much trouble to figure out which choice is equivalent to "Network and Sharing" if you're using Windows XP or Windows Vista rather than Windows 7 (if those previous versions of Windows call "Network and Sharing" by some other name).
I have no idea how you would accomplish this same feat in Windows 8, if you're using the Windows 8 operating system on your computer.
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Also, just for kicks and giggles, make sure that your Ethernet cable to the computer with which you are having trouble is plugged into the first unused socket. That is, the wireless router should be plugged into the first available socket on your cable modem, and the computer having connectivity issues should be plugged into the very next available socket on your cable modem.