Assess the damage closely. Buff away white rings or smudge marks with toothpaste on a clean cloth or No. 0000 steel wool. Deeper grooved scratches, burns or gouge marks require filling with a wax or shellac stick. Simple scratches are buffed away using various oils.
Clean the entire area before buffing away the scratch. Use a mixture of 2 tablespoons gum turpentine and 4 tablespoons boiled linseed oil in a gallon of warm water to clean. Alternative solutions are twenty percent household ammonia mixed with water or a solution of paint thinner (mineral spirits). Use rubber gloves with each of these cleaners and work in a well-ventilated room.
Buff away surface scratches using No.0000 steel wool. Polish with boiled linseed oil, lemon oil, olive oil or vegetable oil. Use a shellac stick or colored wax crayon-type stick to repair larger scratches or grooves. The scratched area is then buffed lightly with steel wool and polished with oil.
Use the French polish method as an alternative. A tightly rolled, palm-size pad of soft cotton cloth is dipped in shellac. Buff repeatedly in circular motions. Allow each layer to dry before re-buffing. Each swipe with the pad leaves a thin layer of shellac that builds into a polished, glossy finish.