Historically, cavalry was divided into light and heavy cavalry. The difference between them was primarily how much armour is worn by the soldiers, and thus how powerful their mounts had to be in order to sustain the burden.
Before the Iron Age, the role of cavalry on the battlefield was largely performed by light chariots. The power of mobility given by mounted units was recognised early on, but was offset by the difficulty of raising large forces and by the inability of horses (then mostly small) to carry heavy armour. In the armies of the Greeks and Romans it played a relatively small role, but was more important in the countries to the east, and horse archers were dominant on the vast Russian steppes for thousands of years.
The decline of Roman infrastructure made it more and more difficult to field large infantry forces, and during the second and third centuries cavalry began to take a more dominant role on the battlefield, also in part made possible by the appearance of new larger breeds of horses. The appearance of the stirrup allowed for the appearance of the heavy mounted knights, which were employed as shock troops, whereas earlier cavalry had to be consigned to the flanks. Knights remained dominant in western Europe until the rise of pikemen and longbowmen, and then musketeers, relegated cavalry again to a supporting role. In eastern Europe cavalry remained important much longer and dominated the battlefield until the early 1600s, because of long distances and better tactics. From time to time eastern cavalry unit was meeting western army, usually defeating the latter. After defeat usually westerners were quickly adopting eastern cavalry tactics - one of most famous examples is Gustavus Adolphus.
See also: Military tactics