The word nightmare contains no horse. The mare of the old term was a crushing spirit, the mara of Germanic folklore, that sat on the sleeper's chest by night, and the false association with horses came later by sound alone. The etymology matters to this entry because horse and night terror have been joined in the popular mind by a word that never meant to join them, and dream dictionaries still occasionally repeat the confusion as lore.
The horse's genuine symbolic employment is grander. It carried the sun across several mythologies, from the chariot of Helios to the horses of the Vedic Surya; Odin's eight legged Sleipnir bears his rider between worlds; the winged Pegasus springs from the Gorgon's blood and strikes the Muses' fountain from the rock. Power serving, and sometimes unseating, its rider is the constant, and the old handbooks read dream horses accordingly, by whether the animal is bridled, wild, ridden or thrown from.
Psychology kept the horse close. Freud's most famous child case, little Hans, turned on a boy's terror of horses, which the theory read as the father displaced, and the Jungian tradition reads the dream horse as the body and its vitality, the animal strength that carries the rider where consciousness directs, or bolts. Popular dictionaries compress the lot to freedom and drive. The readings are the traditions' own, and the state of the dream horse, willing, wild or lamed, is the detail all of them saddle first.