Deep scratches cannot be fully repaired at home. These methods may help light surface scratches only.
Paper towels, tissues, shirts, Windex, acetone, banana, peanut butter, metal polish (Brasso).
If the game installs to HDD (“Install Game” in dashboard), the 360 only reads the disc for verification at launch. A lightly scratched disc may install successfully even if it won't play directly. Install to HDD first.
If unreadable: Buy a replacement disc (often $5–$15 used) or digital version if on Marketplace (rare for 360).