You cannot play DS games with the standard SD card; the SD card is too big.
However, if you have a flash card (R4, SuperCard), you can store .nds games in your micro SD card. The flash card will be able to read the .nds game files and you will be able DS games.
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I have just been lookin on the Nintendo website and you can play your DSiware games straight from the SD Card + you can copy DSiware game save files to your SD card. You can't however transfer your DS games to an SD card or transfer your DS game save files to your SD.
You don't need any additional software, just purchase and download games from the DSi shop.
an sd card is a card that stores things it doesnt matter what brand it is
The DSi can't execute code on the SD card, all it can do is store pictures on it, or play music. There's no way to play a homebrew game off the SD card, the DSi simply wasn't made with the means to do so.
The original DS/lite does not use an SD card. You need either a DSi, XL or 3DS, and these use regular SD cards that can be purchased from supermarkets, computer or camera stores.