A double shine (or multi shine, triple shine, etc.) is the repeated use of Reflector as an attack by either Fox or Falco in Super Smash Bros. Melee. It is performed by jump-cancelling a Reflector and using a second Reflector just after leaving the ground, timed so that Fox or Falco will quickly land and allow the pattern to be repeated. It is a very difficult and technically demanding technique and is often practiced at slower speeds in training mode, though it is easier to perform with Falco as his jump squat animation is 2 frames longer. Performing more than two shines in a row in this way is known as "multi shining".
Note that frames given assume that inputs are given during the first frame of every range (i.e. all inputs are frame-perfect). It also assumes none of the shines hit (i.e. does not count freeze frames).