How could there be something wrong with the WiFi card when he can surf the internet everytime he sits next to the computer? Could the signal coming from therouter be too weak for him to pick it up elsewhere? Can I make the signal stronger somehow?

usually it's the card that's too weak. can be the router as well, but not quite likely

Well, we've still had no luck at all with this. In fact, when he could browse the internet just by sitting at the table behind me, now he can't do that anymore. And when I go into the Network and Sharing Center, there is a red X between the router and the Internet. When I click on the X to identify the problem, it says that "There may be a problem with the Domain Name Server (DNS). Windows attempted to find the well known host www.microsoft.com using DNS. The server responded that the name was unknown."

The notebook can still connect to the router, but, it just cannot connect to the internet. Why?

If you need more information about what I have here I have SBC Yahoo DSL service, a Speedstream 5100 modem, and the Toshiba is a Satellite laptop.