Hi, thank you for your question. There are a number of things going on here, but I'll be brief and suggest that you perform a "power cycle" of the DSL router just to kick things off and see if you can connect from the home computer after the power cycle. Just unplug the power connector from the device and count "one, two, three.." Up until thirty seconds, then plug it back in. Let it startup and wait for all the lights to turn green.
Next, perform a "restart" of the home computer. Can you connect then? Try to find out what the issue is. Are there any other accounts on this computer? If not, you could try to create an account and login. Can you connect via this other account? This is important step, to try to figure out whether the problem is limited to this account.
WHAT HAPPENED?
Did something change on the home computer? What were you doing when the problem started? Did you run upgrades/updates or software patches? Which OS are you using? What software were you using when you first noticed the problem? Are you using Windows or Mac OS? What kind of router are you using? Is there a technical support number?
"The access point and device have different SSIDs"
The other thing, based on my research that I conducted to answer your question, is the SSID match. The SSID on the home computer must match the SSID of the router or you won't get a network connect. (LaPorte, 513).
"The settings for WEP (Wireless Equivalent Privacy) or WPA (WiFi Protected Access) aren't configured correctly."
Another possibility (though less likely) is that there is some type of security, such as WEP (Wireless Equivlancy Privacy) or WPA (WiFi Protected Access), the idea being that the router contains a list of approved computers, and the home computer might not be on that list. The ACL is based on MAC addresses and is designed to include every computer that is on this list and exclude all others. (Laporte, 515)
"The gateway, base station or router is incorrectly positioned."
Is the home computer in range? Have you ever been able to connect wirelessly to the internet from the home comptuer? Does the home computer indeed have the necessary hardware (wireless adapter)? Could this be a "range issue or interference"? (Laporte, 515)
"The connection to the wireless access point or router is poor."
To quote Laporte and Soper, "The wireless access point or oruter is responsible for providing an IP address to each wireless client. To see the status of your connect with Windows XP, right-click the wireless icon in the system tray and select Status. The General tabe ...shows the duration of your connection, signal strength, activity, and connection speed. .. To see your computer's IP address, click the Support tabe shown in Figure 8.28. If there's a problem with the connection, your client will have an invalid IP address (0.0.0.0) or will have a privately assigned IP address..."
One way of sorting out what type of problem the home computer is having is, to connect using an Ethernet cable from the router to the home computer. The 10/100/1000 Base-T Ethernet connection via cable would isolate the problem to the wireless connection and verify that there may be a configuration error in the software.
Normallly, you would get a DHCP server (your ISP's wireless router) to issue an IP address from a range of available addresses assigned to your home computer, but it is doubtful that an IP address has been assigned to your home computer.
There are other possiblities, mostly focusing on the probability that the computer is simply not set up properly, I refer you back to your ISP, which has all the details about your connection situation, they would be the best ones to help to resolve this issue for you.
If you'd like to learn more, Leo LaPorte's "PC Help Desk" is an excellent resource that really helps you to troubleshoot problem areas and it's an excellent disucssion and comes highly recommended.
I'll include a bibliographical reference below.
Good luck, and thank you for your question