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I specialize primarily in the 12th C, so this answer may differ a little bit in other centuries. The middle ages cover a lot of time, so any other medievalists covering other time periods should totally pop in and add to this.

It is important to preface this discussion with a few key points about warfare in the high middle ages. First, pitched battles were very rare; most elite fighters would only see one or two in their lifetime. Some never fought battles, even though they spent much of their lives engaged in military activities. Secondly, nobles and aristocrats very rarely killed each other in battle. It certainly happened, but it was far more profitable to merely incapacitate or capture an opponent, and ransom them back to their families or use them as hostages in order to enforce a political outcome or trade for your own men. There are certain exceptions, such as at the Battle of Hastings, where it was politically untenable for William the Bastard to leave Harold Godwinson alive. But if we confine ourselves to pitched battles, and even more specifically to the role of mounted combat in battles, we can find a lot of information about medieval tactics in relation to charges on horseback.

There are a couple of points about the Battle of Hastings that I would like to start with. It is important to note that the Anglo-Saxon forces led by Harold Godwinson did not fight on horseback. Mounted combat was not common in Anglo-Saxon England. The constant in-fighting that had created an 'arms-race' of sorts in Normandy which led to the development of mounted combat and stone castles, did not happen in Anglo-Saxon England to nearly the same extent. The Battle of Hastings began on the morning of 14 October 1066. The English had spent the night, according to William of Malmesbury, "drinking and singing" while the Normans had spent the night confessing their sins and praying. Henry of Huntingdon describes the start of the battle beginning with a speech given by William:

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"Duke William had not yet concluded his speech when all his men, boiling with unbelievable anger, charged forward in their lines with indescribable force against the enemy, and left the duke alone, speaking to himself. Shortly before the warriors entered the battle, one man, named Taillefer, played a game of juggling swords in front of the English nation, and while they all gazed at him in astonishment, he killed an English standard-bearer. He repeated this, a second time. The third time, in the act of doing it again, he was himself killed, and the lines fell upon him.

Then began death-bearing clouds of arrows. There followed the thunder of blows. The clash of helmets and swords produced dancing sparks. Harold had placed all his people very closely in a single line, constructing a sort of castle with them, so that they were impregnable to the Normans. So Duke William instructed his people to simulate flight, but as they fled they came to a large ditch, cunningly hidden. A great number of them fell and were trampled. While the English were continuing in pursuit, the principal line of Normans broke through the central company of the English. When those who were pursuing saw this, they were obliged to return over the said ditch, and the greater part of them perished there... Twenty of the most valiant knights gave their word to one another that they would break through the English line and snatch away the royal banner, which is called the 'Standard'. When they went to do this, several of them were killed. But some of them, making a way with their swords, carried off the Standard."

There's a lot we can unpack here about charging into battle. Philippe Contamine emphasizes the importance of publicly displaying courage while simultaneously minimizing risk, writing "courage was conceived above all as an aristocratic, noble form of behaviour, linked to race, blood and lineage, and as an individual trait arising from ambition and desire for temporal goods, honour, glory and posthumous renown. It was necessary to avoid 'shame' which was engendered... by 'despicable actions', 'cowardice' and laziness." However, he also emphasizes that "the notions of sacrifice and absolute devotion seem alien to medieval mentality". From this we can glean that the nobles at Hastings were eager to charge the enemy and initiate the battle as a means of proving their courage and prowess in battle.

This is consistent of popular literature of the time as well. William of Malmesbury relates that in the hours before the battle, the Normans listened to the Song of Roland, " that the warlike example of that man might stimulate the soldiers ." He does not specify if there were particular sections of the Song of Roland that were sung before the battle, as it seems unlikely that it was sung in its entirety considering the length. However, Roland exemplifies the knightly qualities of courage and vigor in battle. When faced with an ambush by Iberian Saracens, Roland says, "Damn the heart that turns coward in the breast! We shall make a stand in this place, the first blow and the first cut will be ours."

However, we can also see in Henry of Huntingdon's passage about Hastings that mounted charges were not simply an opening move that led to a chaotic battlefield as we often see in film. The Normans performed a coordinated feint in order to create a break in the English lines, which would make a cavalry charge more effective. Orderic Vitalis states that the Normans feinted twice more, but other sources are more vague. Henry of Huntingdon gives us another example of a mounted cavalry charge used in a mixed field at Tinchebrai in 1106:

"When the Breton cavalry (for both the king [Henry I] and the duke and other lines were on foot, in order to fight with more stability) suddenly rushed on the duke [of Normandy]'s line from the other side, and split it open, the duke's host, overwhelmed by such massive strength, was scattered and defeated."

These and other examples tell us that mounted charges were known to be most effective only after a vulnerability had already been created in an enemy force, or when the enemy can be surprised by a charge. But let's look at a couple more examples in which both sides are using cavalry.