Cavalry tactics
History: 

Chariots: Chariot tactics had been the basis for using the horse in war. The chariot's advantage of speed was outdone by the agility of riding on horseback. The ability of horsemen to pass more difficult terrain was also crucial to this change. Horsemen supplanted most light chariots. In Celtic warfare, light chariots (essedum) persisted among mounted troops, for their ability to transport heavily armoured warriors and as mobile command platforms.

War elephants: Elephant cavalry first appeared three thousand years ago, simultaneously in India's Vedic Civilization and in China. Female Asian elephants were used, sometimes in small groups, sometimes in vast regiments of thousands of animals in the 13th century, primarily to produce a tactical "shock and awe" effect in the field. In addition, the large animals provided elevated platforms from which archers could rain down arrows on the enemy, and from which generals could survey the battle.
The psychological effect of war elephants was often their main tactical use. After encountering elephant cavalry in the Battle of the Hydaspes River, Alexander the Great's troops mutinied and refused to press further into India. However, the animals were often not tractable in battle, and when faced with determined opponents, would often flee and trample their own infantry in their flight.
Horse cavalry developed tent pegging tactics to deal with elephant cavalry. If they maintained their nerve in the face of the larger mounts, horse cavalry could rout elephant cavalry, especially by moving into close quarters and attacking the elephants' vulnerable feet. The Mongols would loose arrows at their enemy elephants' feet and legs until the elephants ran and trampled over their own army.

Dromedary and camel cavalry: Next to elephants, camels were the tallest and heaviest animals available for cavalry. They are neither as agile nor as fast as horses. Their use as riding animals, reported from the battle of Qarqar, was more frequent than horses in ancient times. Their advantage was that while they were standing, a mounted archer could aim and shoot with a strong bow from behind an infantry formation. Camels equipped with small cannons gave the Afghan troops an advantage during the third battle of Panipat. Another advantage was their effect on horses, if the horses had never before encountered camels. In the battle of Pterium experienced Lydian cavalry suddenly had to struggle with their horses panicking, when trying to face an attack of dromedary riders. The psychological effect of the best trained and most reliable soldiers being overrun in confusion decided the battle.

Riding and fighting on horseback: At first it was not considered effective to use weapons on horseback, but rather to use the horse as transportation. "Mounted infantry" would ride to battle, and then dismount to fight. For a long time, riders and charioteers worked alongside each other in the cavalry. Early domesticated horses were smaller and shorter than the warhorses of later history. Combined with a lack of developed cavalry tactics and the skittish nature of an untrained horse, fighting on horseback was unintuitive at first.
The first recorded instance of mounted warriors are the mounted archers of the Iranian tribes appearing in Assyrian records from the 9th century BC.
Mongolian troops had a Buryat bow, for showering the enemy with arrows from a safe distance. The aim on horseback was better than in a jiggling chariot, after it was discovered that the best time to shoot was while all the hooves of the horse were in the air. Nevertheless, an archer in a chariot could shoot potentially stronger infantry bows.
Javelins were employed as a powerful ranged weapon by many cavalries. They were easy to handle on horseback. Up to ten javelins would be carried, depending on their size and weight. Thrown javelins have less range than arrows, but often prevailed in use nevertheless, due to its greater mass and armour-piercing capability, usually inflicted fatal wounds more frequently than arrows. Usage is reported for both light cavalry and heavy cavalry, for example, by the Numidia and Mongol light cavalry, and the Celtic, Cataphract, and Mamluk heavy cavalry. The Celtic horsemen's training was copied by the Roman equites. A significant element learned from the Celts was turning on horseback to throw javelins backwards, similar to the Parthian shot with bow and arrows.
Stirrups and spurs improved the ability of riders to act fast and securely in melées and manoeuvres demanding agility of the horse, but their employment was not unquestioned; ancient shock cavalry performed quite satisfactorily without them.  Modern historical reenactors have shown that neither the stirrup nor the saddle are strictly necessary for the effective use of the couched lance, refuting a previously widely held belief.  Free movement of the rider on horseback were highly esteemed for light cavalry to shoot and fight in all directions, and contemporaries regarded stirrups and spurs as inhibiting for this purpose. Andalusian light cavalry refused to employ them until the 12th century, nor were they used by the Baltic turcopoles of the Teutonic Order in the battle of Legnica (1241).
An example of combined arms and the efficiency of cavalry forces were the Medieval Mongols. Important for their horse archery was the use of stirrups for the archer to stand while shooting. This new position enabled them to use larger and stronger cavalry bows than the enemy.

Tactics of light cavalry using bows: Armies of horse archers could cover enemy troops with arrows from a distance and never had to engage in close combat. Slower enemies without effective long range weapons often had no chance against them. It was in this manner that the cavalry of the Parthian Empire destroyed the troops of Marcus Licinius Crassus (53 BC) in the Battle of Carrhae. During their raids in Central and Western Europe during the 9th and 10th centuries, Magyar mounted archers spread terror in West Francia and East Francia; a prayer from Modena pleads de sagittis Hungarorum libera nos, domine ("From the arrows of the Hungarians, deliver us, Lord")
Another fairly popular tactic was known as "shower shooting".
The Sassanid Persians and the Mamluks were the chief proponents of the idea, although Muslim cavalry in India had also been known to use it in battle. It involved a line of fairly well-armoured cavalrymen (often on armoured horses) standing in a massed static position, or advancing in an ordered formation at the walk while loosing their arrows as quickly as possible. It was very effective against unsteady enemies who could easily be unnerved by the sight of a vast cloud of arrows raining down upon them; however, an enemy provided with good armour and discipline would often be able to hold out at least temporarily against the barrage. A case in point is Procopius's accounts of Belisarius's wars against the Sassanids where he states how the Byzantine cavalry engaged in massed archery duels against their Persian counterparts. The Persians loosed their arrows with far greater frequency, but as their bows were much weaker, they did not do much damage compared to the stronger Roman bows.
The great weakness of mounted archers was their need of space and their light equipment (compared to contemporary heavy cavalry). If they were forced to fight in close combat against better armoured enemies, they usually lost. Furthermore, they were not suited for participating in sieges. For example, although victorious in the field the Mongols originally had been unable to take the fortified Chinese cities until they managed to capture and enlist the services of Islamic siege engineers. The Mongols subsequently failed to retake Hungary in 1280 after the Hungarians became more focused on Western European heavy cavalry and castle building. Good cavalry troops needed much training and very good horses. Many peoples who engaged in this form of classical cavalry, such as the Hungarians and Mongols, practically lived on horseback.
The Battle of Dorylaeum (1097) during the First Crusade shows the advantages and disadvantages of mounted archers; the rider groups of the Seljuk sultan, Kilij Arslan I, were able to surround an army of Crusaders and shoot them from a distance. Suddenly, reinforcements under the command of Godfrey of Bouillon arrived, and the Seljuks themselves were encircled. They could no longer escape and were annihilated in close combat. The defeat of the Seljuks at Dorylaeum was so complete that the Crusaders then crossed Anatolia virtually unchallenged.

Tactics of heavy cavalry using lances: Medieval European knights attacked in several different ways, implementing shock tactics if possible, but always in formations of several knights, not individually. For defense and mêlée a formation of horsemen was as tight as possible next to each other in a line. This prevented their enemy from charging, and also from surrounding them individually. The most devastating charging method was to ride in a looser formation fast into attack. This attack was often protected by simultaneous or shortly preceding ranged attacks of archers or crossbowmen. The attack began from a distance of about 350 m (1,150 ft) and took about 15–20 seconds to cross the contemporary long range weapon's effective distance. A most important element, and one not easily mastered, was to stay in one line with fixed spaces while accelerating and having the maximum speed at impact.