A double shine (or triple shine, and so forth, as it can be used an infinite amount of times) is the repeated use of Reflector by either Fox or Falco in Super Smash Bros. Melee. It is performed by first using Reflector, jump-canceling it, and then shining again on the first frame that you are airborne. It is very hard to perform and is often practiced at slower speeds in Training mode, though it is considered easier to perform with Falco due to his longer jump animation.
FreddyP from GameFAQs says of double shine, "Double, triple, quadruple, and so on shine is just shine, jump, shine, jump, shine, etc."
It is performed by first using Fox or Falco's Down B, jumping out of the shine (using X or Y, not up on the control stick), then using Down B again, on the first frame you are in the air. If done correctly, it shouldn't look like your character ever left the ground. For Fox, you should shine 4 frames after your jump, and For Falco 6. This is a very difficult thing to do (it's a one frame trick), so you shouldn't expect to be able to do more than a double or triple shine consistently.
This technique does not have very many uses outside of looking flashy. Most players only do it to show off their tech skill. It does have a few uses though, which are applying shield pressure, and using it on an opponent pushed up against a wall to trap them there, while doing percent.