Who set up the wireless router? It sounds like they set a security passphrase ?
If the router has been supplied by your internet provider they may have writen the passphrase on the base of the router or listed it among the paperwork.
If it's a recent model it may have been set to WPA (which the Nintendo may not support).
You can check this (and probably the passphrase) by going to the networking utilities on the computer which is successfully connecting to the router.
If all else fails, download the user manual for the router from Linksys and read up on how to address the router's user interface and change the security settings.