A double bridle, also called a full bridle or Weymouth bridle, is a bridle

Bridle

A bridle is a piece of equipment used to direct a horse. As defined in the Oxford English Dictionary, the "bridle" includes both the headstall that holds a bit that goes in the mouth of a horse, and the reins that are attached to the bit....

 that has two bits

Bit (horse)

A bit is a type of horse tack used in equestrian activities, usually made of metal or a synthetic material, and is placed in the mouth of a horse or other equid and assists a rider in communicating with the animal. It rests on the bars of the mouth in an interdental region where there are no teeth...

 and four reins (sometimes called "double reins"). One bit is the bradoon (or bridoon), is a modified snaffle bit that is smaller in diameter and has smaller bit ring

Bit ring

The bit ring is the ring on the side of a horse's bit, particularly on a snaffle bit. It is used as a point of attachment for the cheekpieces of the bridle and for the reins. It also has an effect on the action of the bit...

s than a traditional snaffle, and it is adjusted so that it sits above and behind the other bit, a curb bit

Curb bit

A curb bit is a type of bit used for riding horses that uses lever action. It includes the pelham bit and the Weymouth curb along with the traditional "curb bit" used mainly by Western riders....

. Another term for this combination of curb and snaffle bit is a "bit and bradoon," where the word "bit" in this particular context refers to the curb.
Double bridles are most commonly associated with dressage

Dressage

Dressage is a competitive equestrian sport, defined by the International Equestrian Federation as "the highest expression of horse training." Competitions are held at all levels from amateur to the World Equestrian Games...

 and certain horse show

Horse show

A Horse show is a judged exhibition of horses and ponies. Many different horse breeds and equestrian disciplines hold competitions worldwide, from local to the international levels. Most horse shows run from one to three days, sometimes longer for major, all-breed events or national and...

 classes where formal tack, attire and turnout is standard.