Let me try to help here. please pay careful attention when reading as this may get a little complicated. A router is a piece of networking equipment that allows 2 or more networks to talk to eachother. Inherantly and originally, routers did not have Wi-Fi funtionality in them. Today, most routers are sold as a dual model of router and wireless access point. When your DS is looking for a Wi-Fi connection it is not looking for a router, it is looking for an access point. As said, access points tend to be part of routers now, but it is an important thing to remember that they are not always one and the same.
shady1415, it sounds like your problem is that your modem has a built in router, but not a built in wireless access point. What you'll likely need to do is either pick up the Nintendo Wi-Fi USB adapter or pick up a wireless access point to get your network running Wi-Fi.
Tetsusaigas WindScar!, Your post seems to reference to the USB charging cable you can occasionally find. this will not help you in any way in an attempt to get your DS online. I'm not sure I understand the bit about having Wi-Fi but not having wireless broadband. Does this mean you have an existing Wi-Fi access point in your home connected to a broadband internet connection? There is wireless internet service, but that's not quite what we're discussing here. if you can elaborate I can try to help you further.