The Nintendo DS is a handheld console produced by Nintendo in 2004/2005. The main selling point was the use of dual screens for gameplay, with one being a touchscreen. It is the only console to have overtaken the PS2 in lifetime sales, as a result of attracting a large amount of casual players, and even non-gamers, into the gaming community.
Emulating the DS has proven to be a pain for emulator authors due to its rather unusual hardware, although in some cases bugs that occur in emulators are due to the games being poorly-optimized or programmed.
DeSmuMe is very good, and very well developed but works best with higher end computers. This emulator aims for accuracy over speed but you can tinker with the vast amount of setting to get some extra FPS (see Common Problems and Solutions for tips). If you're still having a hard time running anything without the output playing like syrup try no$gba. DeSmuMe is recommended regardless of your PC power. Desmume isn't perfect or fully accurate so you can expect bugs in some games.
No$GBA focuses on speed and is written in x86 assembly language but it's closed source and rarely updated so it can have compatibility issues and glitches in some games. As this was first a GBA emulator, the 3D added by the DS is still very poorly handled. However, it might be an option for a very low end machine but don't expect a lot of games to run perfectly, or at all. A fan program, no$zoomer, was released for version 2.6 which increases compatibility and more setting, as well as the titular zooming option. the biggest boost is noise cancellation which will clear up the static no$gba makes with 3d rendering. Version 2.7a isn't compatible with no$zoomer yet but does add resizing windows options but any increase of emulating skills is not noticeable. Only use no$gba as a last resort. The only good thing is that it is practically the only DSi emulator.
DS emulation isn't perfect yet so you can expect some bugs.