I have heard that the Norman use of cavalry is what allowed them to conquer England in 1066. Is this true? Did nobody else use cavalry?
Here's something I wrote some time back on the prominence of cavalry in the early Middle Ages and the Normans at Hastings in particular.
Traditionally, and by this I mean in pop history, the supremacy of the knight (or at least heavily armored mounted combatant) in Medieval warfare is dated to one of a few specific times. Sometimes the Battle of Adrianople in 378 is claimed as the start of the dominance of cavalry on the battlefield, the Gothic cavalry routed the Roman cavalry then flanked the Roman infantry and secured one of the greatest defeats in Roman history, but the date that you see most often claimed is the Battle of Hastings in 1066.
At Hastings the infantry army of the English king Harold Godwinson held its ground in the face of repeated charges from the Normans, including the Norman cavalry, but following a series of feigned withdrawals the English infantry was lured out of its defensive posture and weakened in a series of counter charges. Harold King was killed late in the day, either by an arrow to his knee eyeball or hacked down by a cavalryman. This action is traditionally heralded as the beginning of the age of dominance for heavily armed and armored cavalry, eventually becoming the knights we are so familiar with today, that lasted for the next several centuries.
However this analysis does not hold much water upon closer scrutiny. As mentioned above, the role of heavy cavalry in deciding battles is not exclusive to the Middle Ages, Hannibal, Alexander, Fritigern, and others all turned the tides of famous battles through cavalry action long before William was a glimmer in his father's eye. It would deeply problematic to say that William's use of cavalry was the herald of a new form of warfare when the Macedonians, Greeks, and Carthaginians had used cavalry to similar effect centuries before the birth of Christ.
Furthermore the role, and tactical dominance, of cavalry at battles such as Hastings are not as clear cut. Their role in breaking the English lines is often overstated as the English did not break in the repeated feigned retreats, but only fell apart after the death of Harold (it is perhaps possible that the Norman ranged contingent of archers, crossbowmen, and others were the actual decisive factor of the battle, if it is true that Harold died from a missile in his cranium and that the English army retained its integrity up until that point). Nor was the Norman victory a clearly cut rout. The battle lasted nearly all day and repeated charges failed to demonstrate their clear superiority over the English. Indeed, the battle very nearly became a case of a compact and defensive force of infantry overcoming the more "innovative" tactics of the Normans and their cavalry. Obviously though the English did not prevail, however it is not a clear cut case of one side having cavalry, the other not, and this being the deciding factor of the battle.
So this is one part of the argument, the cavalry of the early medieval battlefield were not unstoppable juggernauts that rode roughshod across the battle fields of Western Europe. Indeed there were numerous instances where aggressive infantry action swept away cavalry. Far from dominating the fields of early Medieval Europe, cavalry could indeed be quite vulnerable to aggressive formations of infantry. This was the case at the Battle of Lechfeld, where the Saxon/German infantry of Otto I successfully pushed the Magyar cavalry into the river. Indeed Bernard Bachrach goes through a list of battles supposedly dominated by cavalry in the early Middle Ages and demonstrates that no such evidence of cavalry superiority can be found in most cases.
However the second part of this question is an issue of training. The implication that I see here is that the infantry of the Middle Ages were far too undisciplined to successfully stand up to cavalry. This, quite frankly, is hogwash for two main reasons.
Guy Halsall makes this a focus point of his Warfare and Society in the Barbarian West, now his focus is heavily focused on Francia and Anglo-Saxon England, but his analysis likely holds true for not only western Europe in general but the broader early Middle Ages across the Mediterranean world. He proposes though that the binary of "infantry" and "cavalry" was not an actual feature of early Medieval warfare, and that the warriors in most of these societies would have been deployed as either depending on specific circumstances. He finds the idea that the elite of the early Medieval world were so specialized at this point in time lacking in evidence. The elites of society would have been familiar with fighting both mounted and dismounted.
Now there are some limitations to this argument. The prominence and role of cavalry in Scandinavian and Anglo-Saxon/English forces in the early Middle Ages is contentious. Halsall argues from limited pictographical evidence that the Anglo-Saxons kingdom of Northumbria for example fielded cavalry against the Picts. (Personally I find his conclusions a bit too far reaching, and there are several instances where Halsall seems to conclude that because elite warriors in society rode horses they necessarily had to fight on them as well) and extends this to a broad geographic swathe of land.
Now lets get to the second aspect. There is a common assumption that elite warriors need training to reach that status. This is something that is very ingrained these days. We imagine the best of the best warriors of the ancient world as analogues to our modern special forces and that necessitates massive amounts of intensive training. Only....that's not how these martial societies worked. Knights were not the military elite because they trained the hardest and had the fanciest equipment. They were the military elite because they were a societal elite. Knights, huscarls, thegns, etc... did not wake up each day and run miles, lift weights, drill with their comrades, etc... they were social elites, they had social connections to maintain with their superiors and retainers, they had business to conduct, obligations to meet, hunts to go on, etc... This is true of both warriors who fought on horseback or on foot. The training regimens of infantry warriors were the same as cavalry training regimens, which is to say that they did not really exist.