"My Xbox 360 is on the 1st floor and my router is upstairs on the 2nd floor of my house. What are my options for connecting my Xbox 360 to my network?"
If your router is wireless, you could simply get the Xbox 360 Wireless Network Adapter and connect (assuming you have a decent wireless network signal between floors). I'm in a similar situation. My Xbox 360 is upstairs and my router is downstairs. For a long time, I've used the Xbox 360 Wireless Network Adapter to play games and stream video to my Xbox. Sometime recently (possibly after my wife got her iPhone), the wireless network seemed to hit a point where too much traffic was causing hiccups in streaming video. I recently switched to using an Ethernet over Powerline solution, which has solved all the problems with my video streaming. Here's a video showing how Ethernet over Powerline works:
You can find affordable solutions for Ethernet over Powerline on ebay.
