I have a laptop using a wireless Ralink USB card to connect to the internet. Is it possible to use the unoccupied Ethernet cable slot to hardwire into a router and use the router to send out the wireless signal picked up by the USB card to other devices?
I know I shared my connection like this to my xbox 360 but wasn't sure if it could be used with a router.
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Not exactly. But what you can do is patch the wired network connection to another PC/laptop, or any other wired device, even a switch if you need to support multiple devices, and then BRIDGE the wireless and wired network connections. Now all the devices on the wired network connection have wireless access courtesy of your laptop's wireless network connection (not the router).
Go to Network Connections (type "ncpa.cpl" (no quotes) at any command prompt and hit enter), select both the wireless and wired network connections, right click, and select Bridge Connections. That's it.