wi-fi connection is when you connect to the internet using a wireless router or a nintendo usb connector to play your games against other players from around the world. To use wifi you need broadband and a wirless router.
Ds download play is usually used to play multiplayer games. For example, you have mario kart and if your friend has a ds but doesn't have mario kart then your friend can use ds download play to download a kind of temporary game which you race a shy guy. Also download play can be used to give your friends demo's of certain games. Anything you download to your ds is unfortunatly not permanent as when you turn your ds off it is gone.
On your game mario kart you will not use ds download play, but your friend will. Anyway it is very simple to use download play. I will tell you how to do it on your game. First go to multiplayer on your game and select simple(for players who don't have maio kart). Then your game will search for anyone using download play. At this point your friend must select download play from the main menu. Both ds will search for each other. On your friends screen a bar (like the four bars on the main screen that you can touch) with mario kart written on it will appear. After your friend has touched this thier name should appear on your screen.( if more friends are playing then simply ask them to download aswell) After you are happy that every one has joined then press cut off. Then you are set to race. Hope that helps.