Wireless router that doesn't need a desktop

"I think you're not describing a router but rather a cable modem:

Some cable modems are meant as single-computer devices and expect the login information to be sent from the computer.

The solution in such a case is to attach a router to the cable modem and have the router provide this login information. That way, the router keeps the cable modem connection alive, and it doesn't matter whether any one or more computers are on or off.

What router should you buy:

I would recommend any router that has wireless as well as (at least one) wired port, because the wired port is handy for troubleshooting when wireless fails. In particular, Linksys routers often get positive comments. Here are some that you could consider.

If you dare, there are even custom firmwares for these that enable you to control your internet connection in many good ways - see here for details."