You can use games with an Action Replay DS, but the device doesn't store or load games directly like a flashcart. Here's the breakdown:
Downloading ROMs for games you do not own is illegal in many jurisdictions (copyright infringement). Creating backups of your own cartridges for personal use sits in a legal gray area depending on local laws (DMCA in US).
In short: Action Replay = Cheats. Flashcart + MicroSD = Game Storage/Loading.