Well, every network device has a unique identifier called the MAC address (Media Access Control, not an Apple brand ;-). Your Wii, your iPod, your Xbox, network interface cards in your PCs, and your router all have unique MAC addresses that look similar to this: f0:6d:cx:3d:x9:29
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So, as @dpworkin suggested, connect to the admin panel of your router and bring up the list of connected devices. Depending on your router’s firmware, it should show you a listing of connected devices, their hostnames, their IP addresses, and their MAC addresses.
If you see one of your IP addresses is doubled up with two MAC addresses, then your suspicions were correct. Depending on your router, you might then be able to configure it so that it will block any future connections from the rogue device’s MAC address.
If you’re not able to see a single IP address with more than one MAC address associated, then you should audit the MAC addresses of all your personal devices. Any left over from the list will be your rogue device.