A Nintendo DS flash cartridge is a device like a game cartridge which uses conventional flash memory such as microSD cards to store games, and provides access to them using a menu system. They are normally promoted as a way to run homebrew, but are also often used to run backups, and so have gained a reputation as piracy devices.
Popular examples include R4, its clones, M3, the SuperCard DS duo, and Acekard, among others. They run on special software (often called firmware) which includes the menu. The current revisions of the R4i system software include MoonShell 2, a media player and text viewer.