I believe you're talking about the GBA slot.
The GBA slot (Game Boy Advance) allows the DS to play some GBA videos and single player games that are compatible with both the DS and GBA. There are also a few DS-only games that require the use of the GBA slot for extra game accessories (a music game might have the ability to have a keyboard in the slot).
If you find what appears to be a plastic cartridge in the GBA slot, it is likely the dust protector for when you don't have a GBA game in it. There's no real purpose to it, it just makes the device look more complete, rather than having a hole on the bottom of the system.