It depends on the model of 360. If you're talking about the big white one, it uses a hard drive in a special enclosure that clips onto the top of the system. You can also expand your storage by simply plugging in an external hard drive to the system's USB port.
If you're talking about XBox One, I don't know why you would need a separate, 360-centric hard drive to play 360 games- you just need enough storage space. When you insert a compatible disc, the game downloads to the system's hard drive and then you can play it.