You don't "save" games to the Xbox 360 like files—you install them for faster loading, but the disc or digital license is still required to play.
| Action | Disc Required? | Storage Used |
|---|---|---|
| Install disc game | Yes (to launch) | ~6–8 GB |
| Buy/download digital | No | Full game size |
| Save progress | N/A | Few MB |
Respect copyright—installing discs you own is fine; copying rentals or borrowed discs is not.