I have forever looked at the DS Lite's title screen and felt disappointed about how little it could do. Now, the DSi has everything you ever wanted.
Hardware:
The DSi is a bit longer and a bit thinner, but you wouldn't notice. The matte finish is much better than the glossy finish of the DS Lite, as there will be no more finger prints, but it is prone to becoming dirty. The screens are 0.25 inches bigger, which may not sound like a lot, but it is definitely noticable when you step from the DS lite to the DSi. There is no more GBA slot, but that doesn't really matter. The volume controls are now buttons, which is much better for keeping sound quiet in public areas, but not so much so that YOU can't hear it. There is now a power button on the inside that can also return to the menu. The speakers are better and louder when you want it. There is a 0.3 MP camera next to the microphone on the inside of the hinge and one on the outside. The stylus is longer and fatter. There is an SD Card slot aswell for photos and music. Internally, there is a faster processor which games in the future will utilise.
Software:
The Wii-like menu looks very nice and is easy to use. Game cards can now be hotswapped (changed without having to turn off the system). The DSi Shop allows you to download games or DSiWare, of which there are inceasing numbers all the time, as well as the brilliant, free DSi Web Browser, which is much faster and better than expected, but cannot run Flash unfortunately. When you first log on the the DSi Shop, you get 1,000 free points (offer until march 2010 I think).
The DSi camera is only 0.3 MP, but that is enough for use on the console. There are some fun effects you can add too like Distortion Lens, Colour Pad, Mirror etc. The camera has face recognition for some effects, and there will be DSi exclusive games coming out in the future that use this feature. DSi Sound is another good program, allowing you to record sounds and play with them. More importantly, music on the SD card can be played with some fun equaliser graphics, and be slowed down and speeded up if, say, it was a learn a language CD and you wanted to slow the words down. I takes AAC files which means that you can put on your iTunes collection (but you must convert it in iTunes first). There your old favourites too: Picto Chat and DS Download Play. And remember, there is a huge library of great DS games, that this will still play (except Guitar Hero due to the GBA slot)
Overall:
This is a great console. If you have already have a DS Lite, it is a little expensive to upgrade, and I would wait to the prices come down. If not, i would fully recomend it and say that it is better than the iPod Touch and PSP.