You do not need an inverter for an Xbox 360 at home. Inverters convert DC (battery) power to AC (wall outlet) power. The Xbox 360 runs directly on standard household AC power: 120V (North America) or 230V (Europe/Asia).
You would need a car power inverter (12V DC → 120V/230V AC). However:
Just plug the Xbox 360's included power cord directly into a wall outlet. Use a surge protector for safety. No inverter, no converter, no adapter needed.