Short answers, pulled from the story.
A palfrey was a smooth-gaited riding horse used by medieval nobles, ladies, and high-ranking knights for long-distance travel, hunting, and ceremonial occasions. Knights also rode palfreys to the battlefield to save their heavier warhorses from fatigue before combat.
A fine palfrey could equal the price of a knight's destrier, making it one of the most expensive animals in the medieval world. Its high value reflected both its selective breeding and the heritable genetic trait that produced its smooth ambling gait.
The palfrey used an ambling gait, a smooth four-beat movement that kept only one foot fully off the ground at a time. The trot is a two-beat gait with a moment of suspension between beats that jolts the rider; the amble eliminated that suspension, making long rides far more comfortable at roughly the same speed of about 8 miles per hour.
Ambling horses declined in Europe as roads improved and carriage travel replaced riding for long distances, since carriages required large trotting horses. The rise of the Thoroughbred and light cavalry breeds, which favored the gallop and tended toward trotting gaits, further displaced ambling lines from European breeding programs.
The Paso Fino and the Peruvian Paso, both developed in Latin America, are considered among the closest modern descendants of the medieval palfrey, each performing two or three distinct ambling gaits. Other gaited breeds carrying the tradition include the Missouri Fox Trotter, the Tennessee Walking Horse, the Icelandic horse, and a sub-group within the American Saddlebred.
The word palfrey descends through Old French and Latin from the Gaulish word vorēdos, meaning a post horse or riding horse. The German equivalent, Zelter, is cognate with the Icelandic tölt, the name still used for the distinctive gait of the Icelandic horse.