you've got 2 options for multiplayer. Ad-hoc and Nintendo WFC. some games only require a single cart in Ad Hoc mode, but all games require each of you to have a cart in WFC mode.
some games (such as Mario Kart) allow you to connect and join random games with random people all over the world, who are also interested in playing with people such as yourself. Other games issue you with a friend code that you can give to people you know, and you pair your friend codes and play with each other. Mario Kart offers both these options, whereas games like Animal Crossing only offer the latter
i've got a post in this thread which summarises how to get around one of the main problems people face when connecting to the net with their DS, but the main thing to do it just try it - it's not going to hurt if your DS isn't compatible with your router. if it
isn't compatible, you can get a Nintendo WFC USB dongle to stick in your PC and achieve wirelessness that way. When faced with a problem in-game, i've found it's best to just google the error message you get (sometimes you'll get a code, which is nice and google-friendly

) as there's bound to be people who've, at least, mentioned it before. Nintendo WFC's forum normally has quite good information regarding your problems in, but it's not always that in-depth, which is why i recommend google
