In terms of hardware, there are currently two versions of DS... the older DS (some call them DS Phat) and the newer DS Lite.
In terms of versions, it refer to who distributed the hardware. The so called "local version" refer to DS made for Asian and Australian region (excluding Japan and China), and are distributed by the local distributor, Maxsoft. They come with 3 months warranty. Back in the middle of 2006, lots of gamers complained that the local distributor forced people into buying DS Lite bundled with certain titles at "special price". Incidentally, these games were games that were basically stockpiling in the warehouses. The local distributor also played a number of stunts on the local community... like releasing only the light blue version of DS Phat, instead of the standard 4 colours JP versions.
The local community has spoken out against Maxsoft though. Refer to this petition stating the grievances of the community.
http://www.petitiononline.com/WiiSg/petition.html
Apart from local version, JP version of DS are widely available in Singapore. Mine is a JP version of DS Phat, bought when the 4 colours (Graphite Black, Pink, Light Blue and White) were announced. It was considerably more visible pleasant than the original DS which only comes in Silver.
For DS Lite, the JP versions so far has Pink, White and Black (grey).
There's European versions of DS Lite as well, with White and Black (real black apparently).
The US version so far is only White.
Is there any differences in getting whichever version? As far as I know, apart from the 3 months warranty (which is inconsequential anyway!), nothing at all! Games are not region specific and whichever version would come in built with different languages anyway. The packaging may be different as well, for instance, mine come with a japanese manual. The pricing may be a bit different too, for instance the EU version of DS is slightly more expensive.
Frankly, I won't support the local DS version at all... Local Wii on the other hand is a separate matter. Still pissed of the non-commitment of Maxsoft on Wii though. When is it coming?