The DS is the oldest, shortest battery life and has a screen that is not back lit, it plays DS games and GBA games
The DSlite is the second release from Nintendo, it features a back lit screen, a longer battery life, a smaller form factor and still accepts both DS and GBA games.
The DSi is the third release, back lit screen, slightly longer battery life. The new features included two cameras, one on the inside and on the out, an SD card slot and the ability to play music with a built in MP3 player. The DSi also features the ability to download games from the Nintendo store, and save them to an SD card. Photos and music can be placed on the SD card as well, so they can be placed in a friends DSi or dumped onto a computer and shared with anyone. The DSi and all models that follow it, no longer have a Slot 2 for playing GBA games, but still accept DS games.
The fourth entry, the DSi XL, is exactly the same as the DSi, only is boasts a screen that is 93% larger, battery life, I hear, is rougly the same, as it has a bigger battery, but a larger screen that takes more juice. The large screen is a plus and minus for some people. While newer games will look nice and clean older DS titles will look stretched or blurry. The button layout stays exactly the same, Nintendo didn't think that the spacing of the buttons was an issue and kept the four ABXY buttons the same distance apart.
The last entry to the DS line up is the 3DS, a gimmicky at best attempt with 3D, I would argue that no one should use the 3d settings. The thing gives me a headache in about ten seconds. However, the 3DS boasts better battery life than the DSi and DSiXL both, as far as I'm aware. As far as games go, the 3DS, while having a full selection of its own titles that are very good (even revamping older titles from the n64) it still has access to the entire DS cartridge line up, and the Nintendo store for downloading games. The 3DS also has two cameras (3 technically) with on on the inside and one on the outside that is capable of taking 3D pictures.
As far as accessories go, each model as its own line of doo-dads. Batteries, while accessible by the end-user, are sold with the unit, as is one AC wall charger. I'm sure you could find a battery online, but the things are screwed down inside the DS, making changing to a battery on the go not feasible. Instead, I would recommend a portable charger. These units are available from $20usd and up, and hook up to most devices via USB