Throughness is a term used in equestrianism

Equestrianism

Equestrianism more often known as riding, horseback riding or horse riding refers to the skill of riding, driving, or vaulting with horses...

 to denote an absence of resistance in the horse to the rider's commands.

A 'through' horse is perfectly submissive, allowing the rider’s aids

Riding aids

Riding aids are the cues a rider gives to a horse to communicate what they want the animal to do. Riding aids are broken into the natural aids and the artificial aids.-Natural aids:...

 to go freely through the animal, with the rein

Rein

Reins are items of horse tack, used to direct a horse or other animal used for riding or driving. Reins can be made of leather, nylon, metal, or other materials, and attach to a bridle via either its bit or its noseband.-Use for riding:...

s influencing the forehand, and the riders' seat and legs influencing the hindquarters. When completely through, the horse is soft and elastic, with a connection from back to front, balanced and relaxed. It is supple and attentive to the rider's aids, and will willingly respond at the slightest touch, not only to the driving aids, but also to the restraining aids.

Throughness is often compared to a circuit of energy between horse and rider: the rider's leg aids encourage energetic movement in the hindquarters, which push the back upward, which in turn allows for connection with the front end and the bit, and the connection felt in the bit transmits a feeling of energetic movement back to the rider's hands. Of course, this is a question of "feel", meaning a very soft reaction in the rider's hands. If a rider gives driving aids and the horse responds by putting a lot of weight into the rider's hands, the horse is not "through" at all, but unbalanced and dependent on the hands of the rider to keep itself in balance.

Throughness is most important in 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...

 riding, essential for impulsion

Impulsion

Impulsion is the pushing power of a horse, which comes from its desire to move powerfully forward with energy. However, speed does not create impulsion, and a rushing horse is more likely to be "flat" than impulsive...

, but a through horse can make riding easier in all equestrian disciplines.