Apply toothpaste or Brasso to a clean cloth.
Rub this cloth, using a moderate amount of pressure, onto the badly scratched areas of the CD. While doing so, the toothpaste or Brasso will slowly wear away the top plastic layer of the CD. As the plastic wears away, so will the scratches.
Take a clean cloth and dampen it with rubbing alcohol after most of the scratches have worn away.
Clean the toothpaste or Brasso residue off of the CD using this cloth.
Allow the CD to dry and then test the CD in a computer or CD player.