A double bridle is a bridle with two separate bits and two pairs of reins (as opposed to a Pelham bit, which is a single bit used with two pairs of reins).
That is, a double bridle is a bridle specifically designed to accomodate two bits and two pairs of reins. Most often the two separate bits are a snaffle and a curb bit.
When training your pony in a double bridle, the snaffle bit is used most of time with curb bit used for fine tuning your pony's movement and head carriage. It may take some practice to use two sets of reins.
A bio-horse rider will typically put one set of reins between the pinkie and ring fingers and the second set of reins just above them (between the middle and ring fingers). However, you should use whatever method works best for you.
It is important not to confuse the two pairs of reins since each will have a different action, with the curb bit reins producing a more harsh effect (of course pulling on both sets of reins simultaneously will produce an even more harsh effect and will be hard for even the most recalcitrant pony to resist).
A double bridle can be used in ponyplay (albeit not a horse sized double bridle; several human pony bridles can be used as double bridles though you will often have to add a couple additional straps to attache the second bit). The human mouth can accomodate two bits.
The times when I have worn a double bridle, I have used a ported bit as the curb bit, and a thin mullen mouth metal snaffle (see photo at top right and below). The snaffle bit had to pulled back and down so as not to interfere with the port portion of the curb bit, but both bits fit in my mouth surprisingly comfortably.
A thicker snaffle (e.g. the bit used in most rubber bit gags) actually hinders the effectiveness of the setup in my opinion. Sure, your mouth is stuffed full, but when your trainer applies pressure to the curb reins, the action will be much less severe since (in my experience), the larger snaffle blocks the curb bit to a degree.
The curb bit has a lever action (make sure that the curb chain is secured under the pony's chin to achieve the proper effect) which can be enhanced depending on the mouth piece of this bit. The use of a spoon type or spade mouthpiece on a curb bit can be particularly effective in controlling a pony.
A double bridle is fairly uncommon in pony play, but not unheard of. It is typically used in dressage where (similar to the bio-horse world) nuanced rein cues are necessary to get a pony to move in a specific way.