As luck would have it, I've been playing around with this pretty extensively across the past few months. My advice would be to forget about it- the DS' hardware just isn't fast enough to emulate many things properly. NES (nesds) and Game Boy/Color (gameyob) emulation's pretty much perfect- but anything newer has issues. Furthermore, the DS' screens' resolution is too low to display the full output of systems designed to output to a television properly- meaning the video output's either cropped, or has to be downscaled and smoothed- which looks alright, but very obviously worse than the original output should look.
Mega Drive/Genesis emulation's okay but comes with a lot of caveats. SNES emulation is too unstable/slow/inaccurate to really bother with.
The question of what flashcart to go for for the best emulation experience is, strictly speaking, one better-suited for /r/Flashcarts or GBATemp than here- but there's only really one answer- the Supercard DSTWO. It gets around the DS' hardware limitations by including a significantly faster CPU in the cartridge itself- with certain emulators coded specifically for it. It's the only way to get halfway-decent GBA emulation on the DSi, for example (TempGBA-MOD). Unfortunately, that's about the only thing it is good for, emulation-wise- you can technically run a SNES emulator on it (CatSFC), but it's too slow to be of much use at all. Furthermore, video output from code running on the DSTwo's CPU is capped at 48FPS thanks to the limitations of the DS' card slot's bandwidth- meaning every single GBA game played through it will always visibly skip frames unless you force it to render slower than full speed.
Don't go buying a DSTwo if all you want to do is emulate on your DS Lite- they're pretty expensive, and come with significant downsides beyond that (continual battery drain from the CPU effectively rendering the DS' sleep mode useless, slight audio interference when accessing data from the MicroSD card, not really the best flashcart UI). It's only really useful if you have a DSi or 3DS/New 3DS.
If you really do want to play around with emulation on the DS Lite, pick up an original R4 (cheapest, only supports MicroSD cards up to 2GB), an Acekard 2i (cheap, now it's obsolete- doesn't work on newer DSi and 3DS firmware), or an "R4i Gold 3DS" (best, still pretty affordable). You can find them fairly easily on eBay if you set up a search alert for them. Don't go buying an unsupported flashcart- anything beyond those (and maybe the DSTT) just isn't worth the bother nowadays. There were a lot of terrible DS flashcarts made that just stopped being useful after a few years.