The Nintendo DSi is just an upgraded version of the Nintendo DS and DS Lite. The Nintendo DSi plays the exact same games as the Nintendo DS and DS Lite and only plays DS and DS Lite games. Currently, there aren't going to be any DSi games, only more DS games.
Some games cannot be played on thedsi if they are for the dslite, or regular ds. Like if you see Nintendo ds games that say" available for Nintendo ds lite and Nintendo dsi". Nintendo are working on making games for the dsi too.
No, Game Boy and Game Boy Colour games are not compatible with the Nintendo DS. Game Boy Advance games can, however, be played on the Nintendo DS and Nintendo DS Lite. This second slot was removed with the Nintendo DSi and DSi XL to "shy" people away from the older games.