My experience is only on the DS, but all the emulators you'll find are DS emulators (no one can easily, reliably run DSi code yet). It should be pretty easy to set all of these up...a few need ROMs to be in certain folders (read the readme to be sure) but it's really just a matter of copying them to the SD card with any recent-ish flashcart.
Anyway, there are some pretty good ones. I'd say NesDS for NES games (quite a lot of games work). Despite what Brad says, NES games still look pretty good despite the smaller screen resolution. The scaling is really good.
For SNES I'd say SnemulDS...not that you have many choices there.
A lot of games work pretty well, but keep in mind the DS has to uses some tricks to get games running full speed, so you might have to play with the graphical settings and such to get games to look right. It uses the DS's 2D hardware rather than a software renderer...it's much faster, and many games look fine, but some need to be tweaked...it's not perfect but it's probably the best SNES emulation on the DS will be. Also, interestingly, it uses the ARM7 cpu to emulate the SNES' sound hardware. Some timing sensitive games don't like this, so if you have some games that won't work, that could be why. Also, don't try anything needing an extra chip such as the SFX or SDD1. They won't work.
Like Max says, Lameboy seems to be the best GB/GBC emulator. Tons of games work well on it, and I'd say most are full speed (one example that isn't being Shantae, which isn't quite 100% speed).
For PCEngine/TurboGrafix, I think the only choice would be NitroGrafix. I have yet to try it, but it should be a good piece of work, it's developer Flubba based it on his old PCE GBA emulator, and he certainly knows what he's doing.
For Genesis your only choice is jEnesisDS, thankfully it's a pretty good choice. It even has sound, which as I understand it is quite a feat given the available RAM of the DS. Some games are too large horizontally though (it can scale vertically but not horizontally...you can use L and R to scroll left and right though). A lot of games work pretty well just the same though.
For Master System, DSMasterPlus seems pretty decent, though admittedly I only really used it to play the SMS Ninja Gaiden. S8DS should be good too, it's another one made by Flubba.
There's a lot of others too, like MaracaDS (which emulates some old arcade games like PacMan), NeoDS (which, though it's far from perfect, is astoundingly impressive in how well it emulates the NeoGeo on the DS...a lot of games work well if not full speed. Metal Slug is very very playable on it), and DSx86, which emulates a 286 computer and thus lets you play a surprising number of old DOS games on your DS. It grows by leaps and bounds with every release.
Make sure you check out stuff like DoomDS and the Quake port too. There's also a really good port of ScummVM for those old adventure games like Maniac Mansion.
Anyway, if you have any other questions, feel free to ask.