This will be a long list.
Emulators for:
Atari 2600 - Stella DS (entire 500 game Atari library is only like 2MB)
Colecovision - Coleco DS
Arcade Emulator - Marca DS
Neo Geo - Neo DS (surprisingly good)
NES - nesDS
Gameboy - Lameboy
Genesis - jEnesisDS
Sega Master System - ApprenticeMinusDS
SNES - kind of buggy - SNEmulDS
Homebrew:
DS Organize - PDA, file browser, web browser, media player, text reader...does everything pretty much
A Touch Of War.nds
AmplituDS.nds
Atomix DS.nds
Balloon Fight DS.nds
Beneath A Steel Sky.nds
Closed.nds
Codename Hacker.nds
DiceWars.nds
DS All-Stars.nds
DSDoom.nds
Electro.nds
Eurotunnel.nds
Everlasting Love.nds
FrequenDS.nds
GeoWars.nds
Gioioso.nds
Great Giana Sisters.nds
GrowRPG.nds
Homer & The Lost Donut.nds
Jelly Blocks.nds
Kongregate.nds
LemmingsDS.nds
Magic Touch.nds
Martian DS.nds
Mental Games.nds
New Caisse DS.nds
No Place To Hide.nds
NoizDS.nds
Paddle Battle.nds
Platdude Retro Collection.nds
Platdude X-Mas Edition.nds
Plop Invaders.nds
Pocket Physics.nds
PuzzleManiak.nds
Rainbow Sprite Wars.nds
Rudolph, Steady, Go!.nds
Secret Of Monkey Island.nds
Setsuzoku no Puzzle.nds
Snake DS.nds
Spout EX.nds
SquareDS.nds
Submarine Demo.nds
Super Mario Bros 101.nds
Table Hockey Demo.nds
Tomorrow Hill.nds
Touchdown! The Alien Attack.nds
Twist Alpha.nds
Whee! DS 2.nds
Word Wrap.nds
Wordup!.nds
Worms War Online.nds
WusteroiDS.nds
Zork Trilogy.nds
Bliptracker.nds
cf_bench.nds
Colors.nds
Constellations.nds
Corruption.nds
DiagnoSe.nds
DS2Key.nds
DS2Win.NDS
DSOrganize.nds
MegaMania Commercial.nds
Moonshell.nds
Perpetual.nds
Phidias.nds
Pocket Physics.nds
Poke Teh Bunny.nds
Pyramid Of Doom.nds
Ragdoll Demo.nds
SandScape.nds
Shooting Watch DS.nds
Sniff.JazzBox.nds
The Pawn.nds
Treasures of Gaia.nds
Wire World DS.nds
Evil Santa Attacks.nds
Present Factory.nds
ramming Santa.nds
Scogger.nds
Snowflakes.nds
Snowman Attack.nds
Uummarmiut.nds
Virus DS - Xmass Edition.nds
Xmas Entry.nds
Xmas Gifts.nds
Xmaspede.nds
ASCIIWAR.nds
ChessNET.nds
DualShooter.nds
Electro.nds
Game Melody Oratorio.nds
Legion.nds
MarbleMan.nds
Negative Space.nds
Noob MiniGames.nds
Pong Classic DS.nds
Puzl.nds
Star Wars Battles.nds
Tyrian DS.nds
White Dwarf.nds
A Cup Of Tea.nds
Abstract Shoot.nds
AntareX.nds
AquelaBall.nds
ArchonDS.nds
Arcomage.nds
ASCII Mess.nds
AstroLicker.nds
BeatStorm.nds
Biscuit Romp.nds
Breakout DS.nds
BrickFall.nds
Cave Story DS.nds
Crazy Sheep Game.nds
Cube BreakOut.nds
Cube-X DS.nds
Dropping Skill.nds
DS Racing.nds
DSCubeRubik.nds
DStroy.nds
Every ExtenDS.nds
Explosion Of An Aeroplane.nds
Fsaucer.nds
Fushido.nds
Futuristic Racer.nds
Gea vs COdos.nds
Get Out.nds
GrowRPG.nds
JNKPlat.nds
KnifeOshon.nds
Lapinou Jumps.nds
Light Corridor.nds
LinesCS.nds
Lockjaw.nds
Maouss DS.nds
Meteora - Galactic Battle.nds
Metro Wardive.nds
Momentum Ball.nds
Oh Mummy! Evolution.nds
Oil Panic DS.nds
Omalone.nds
Panik Gastrik.nds
Penguin Football.nds
Perilous Wheels.nds
Petzy Ball.nds
Pingalo.nds
Pong++.nds
Return to Tyrian.nds
River Crossing.nds
RushHour DS.nds
Secret Collect.nds
sfCave.nds
Shoot & Score.nds
Squash DS.nds
Storm The House.nds
Super Mario Galaxies.nds
SuperPong!.nds
Tama Survivor.nds
TetattDS.nds
The Reality Game.nds
Touch Me (I'm Famous).nds
Twist.nds
VirusDS.nds
War Of The Weeds.nds
Warcraft - Tower Defense.nds
Without Escape.nds
World Of Sand.nds
YarrgH!.nds
comicbookds.nds
moonshell.nds
Sorry, I just outputted a list of my NDS homebrew folder on my flashcart with dir /s /o /b *.nds >c:\templist.txt so it's pretty hard to read. Honestly, I just dumped every homebrew I could find about a year ago into there and a lot of it I haven't even looked at. I enjoy playing Secret of Monkey Island (other Lucasarts SCUMM games are available) and Warcraft - Tower Defense. I mainly use the emulators for playing old Atari and Coleco games, etc. DSOrganize is a pretty good file browser and basic webbrowser but I will probably put the NDS Opera browser on my system eventually. My favorite app is probably Comic Book DS which allows me to compress not only comic books but any books that I scan for reading on the fly with multiple zoom levels, etc. but best for things with pictures.
For music (mp3, wav, others), movies (DPG format), and e-reader and all those basic things, I just use moonshell which is integrated with my R4 cart. Most carts come with a built in media player or picture and ebook viewer anyway. I use an R4 cart with a 2GB microSD in slot 1 of the NDS with a Supercard Lite with a 1GB microSD in the slot 2 (GBA slot). The R4 lets me boot to Slot 2 and switch carts. Both are probably out of production now and harder to get. I'm a bit out of the loop lately but the Acekard 2 and Edge cards are good budget ones with all the good features. Ones like CycloDS and Supercard SD1 are the expensive higher end ones. I also have an EzFlash V gba loader/ram expansion/rumble pack cart that I haven't used yet because my supercard is in my slot 2. Theoretically this should allow me to run GBA roms and act as a rumble pack in addition to being a memory expansion for those programs that require like the Opera browser which would be great as DSOrganize's browser only has limited memory to access and thusly cannot display most larger pictures on the web (IE: my avatar on CP might show up on DSOrganize but not my signature).
The easiest place to visit is probably
www.dealextreme.com to see what's currently available. I'm told the actual R4 production stopped about a half ayear ago and the new ones are just clones/bootleg ones and to avoid them because they are unreliable.