Flash carts allow you to play pirated games, homebrew, emulators, and do other things like play .MP3s.
If you're looking to buy one now, and don't want to spend much, you can pick up an Acekard 2i. $18 or less pretty much everywhere, runs on every DS model, and thanks to AKAIO1), runs pretty much every game released within a couple weeks.
If you want to throw money at it, pick up a Supercard DSTWO. It can do SNES emulation, and I hear there's a port of MAME4ALL to the thing too.
The R4 was popular, but its firmware is no longer updated. The alternative Wood R4 firmware is updated weekly to support all games and supports most real and fake R4 cards.
Last updated: June 26th, 2011