Double reins are used with specific bits and/or bridles. Bits typically requiring double reins are: kimberwick, pelham, elevator, and gag. The double bridle requires double reins, one for the snaffle (bridoon), and one for the curb.
Riding with double reins is typically more difficult than riding with single reins and is typically reserved for more experienced riders. Each rein represents a different set of pressure points in the mouth of the horse and typically one set is more severe than the other.
Holding double reins depends on the horse, the instructor, the bit, and the rider. Could there be more variables? USPC suggests in the Volume 3 Manual of Horsemanship, that the snaffle rein be held under the pinky of each hand and the curb rein be held between the pinky and third finger.
There are good arguments, however, for holding the snaffle rein where it is without a curb rein, between the pinky and third and holding the curb rein either between the middle and third or under the pinky. The difference is how much leverage needs to be applied to the horse which depends on the horse and the skill of the rider. More leverage is available to the rein held under the pinky.
Old military experts held the reins in other configurations as well including holding all four reins in one hand to allow the other hand to hold a sword which requires good control of the position of the hand holding all the reins. Another configuration is to hold both curb reins in one hand with one snaffle rein in the normal position between the third and pinky with the other hand holding the other snaffle rein. This configuration is used but it is not readily apparent what the advantage is.