The original DS (typically called DS phat) is the original.
The DS Lite is a nicer design, thinner, and weighs less. The GBA slot is modified so that custom addon cartridiges can fit flush in the device. Also there's different screen brightness options, rather than screen on or screen off. I think the build quality feels higher, but I'm not sure. No actual hardware changes, only cosmetic.
The DSi is an upgraded version of the DS. It has double clock speed in supported games, more RAM, a camera, an internal drive, and (I think) SD Card support. You can (not anymore, store is shut down) download DSiWare applications, which are smaller games installed on the device instead of cartridge. The list of physical and digital games are not the same. Some DS games will take advantage of the DSi hardware even though they're not natively DSi game.
The 3DS is the next generation handheld console with significant improvements. It has backwards compatibility with the DS/DSi, and GBA through hacking (No GBA backwards compatibility on stock console unless you bought when it first came out).