As far as I know, the DS's wifi can be used to play certain games online (world wide), local multiplayer, to download demos from other ds's, and for some home brew. To set it up on your DS, make sure you are in a wireless hot spot that is unsecured, or has WEP encryption, the DS will not be able to connect to a wifi signal that is encrypted with WPA or higher. Then you have to have a game that has online features, then go into wifi settings and scan for access points, find yours, put in the key if there is any (everything else can be set to auto) then tap connect. The DS will try to connect, if it does you're set, it might not be able to, in this case, my only thought would be to turn on DMZ in your Router's settings, and make sure that MAC filtering is off as well.
As far as downloading demos, certain DS games allow you to send a demo of the game to a friend, while other cartridges are only demo cartidges that hold several demos and broadcast them all for download(the auctual cartridges do not have any playable content). This is also what they have in the stores. If you download a demo, you can play it as long as your DS remains on, you can even sleep your DS and play upon wakeup, but you need a cartridge in Slot 1 for sleep to work.
Hope I did'nt confuse you.