DS is, well, your basic DS. It can play DS games... and that's about it. Oh, it also plays GBA games. The other systems can't.
DSi is DS + internet browser + camera. It also has downloadable DSi games (almost all require purchase of DSi points).
3DS is DSi + 3D capabilities. It also has a lot more downloadable games, including almost all DSi games, and some included games and videos for free. In addition to that, it plays both 3DS and DS games. And it has some other features like tilt sensors.
New DS games are still coming out, but the releases are getting less and less. 3DS games require a 3DS, and as time goes on, most likely DS games will stop being released in favor of 3DS games.
I hope that clears up everything for you.