The router shows it is connected, but I do not have internet acess

Not sure if you try to set up a WIFI Access Point or are you trying to share internet access through a PC? I will try to answer anyway... You need to connect the RJ45 (that square-ish) to your wireless router on the WAN port.

What you're missing is that ALL the PCs will be connecting THROUGH the router to the Modem. Unplug the PC network Cable from the modem, then plug a network cable from the router's INTERNET port into the modem (where you removed the PC's network cable). Now, plug the PC's network cable into one of the ports on the router (usually 4 or more, marked 1,2,3,4. Think of a Router as a "splitter". It connects to the Modem, then several devices connect to the Router, some of them by wire and some by Wi-Fi. So your desktop PC is connected by wire to the router and your laptop PC is probably connected by Wi-Fi (wire is an option, as well). The Router sorts out all those signals and communicates with the Modem. Sound reasonable?

this is because your modem is out dated your better off getting a new one

connect the router wan port to one of the modems lan ports,
then connect to the router per lan or wlan and everything works.

1. check the line has internet service enable (telephone provider),
2. check the microfilters on each telephone plugs
3. check the dsl light stays always on. if not is a line/provider service problem.
4. enter the router webpage.
5. configure the user and password internet acess codes. (check the internet provider for your internet account)
6. check the internet light turns on

click on start >> go to control panel >> double click on network and internet connections >> right click on wireless network connection >> click on properties and go to wireless tab >> check the box at the top that says use windows utility >> click on ok and than right click on wireless network connection and click view available wireless networks. It will show you your network name. click and connect to your network.

if it still fails than login to your router and change wireless settings

When establishing the connection try placing your psp near the router

To watch videos online from Youtube from you iPod Touch, you must be connected to an active internet connection via wi-fi.
To connect to a wi-fi access hotspot, Go to Settings>Wi-Fi. Now, look under Choose a Network':
-If there is a connection showing, then touch the network name to connect to it. After, retry Youtube and your problem should be solved.
-If there aren't any connections showing, then you do not have a acess point nearby.
-If you do not have a acess point nearby, you can always setup one at your very own home with a router.

I have seen this many times before, and its either the driver or the wireless card itself.
If its the driver, try to delete the driver and then reeboot, your OS will install it again and you then have to set it up for your Router again, if the problem is still the same, you have to turn it upside down and get a new wireless card(its reasenble easy to change).
/Teis
If this helped, do vote :-)

to make sure its setup properly you should reset it up, to do this press the reset button on the back of router, and then when its up to a ready state acess it through your web browser. the adress is 192.168.2.1 ,then click the login button, the defualt password is blank. after that your are logged into the router you can run the setup wizard located on the left hand side of the page. once setup is complete you should be able to acess the internet.