The original Pokémon Red (Game Boy) cannot be played directly on any Nintendo DS model. The DS and DS Lite have a Game Boy Advance slot (Slot-2), but it only accepts GBA cartridges—Game Boy and Game Boy Color carts physically don’t fit. The DS also lacks the Sharp LR35902 CPU and video hardware needed for GB/GBC games. The DSi and 3DS removed Slot-2 entirely, eliminating even GBA backward compatibility.
PC and mobile emulators like mGBA and SameBoy exist, but legality varies by jurisdiction. For ethical emulation, you should own the original cartridge.