I wrote about this topic at length in this thread. At the time the Chigi vase was made, c.650 BC, Greek warfare was insufficiently organised to allow us to identify 'true cavalry' - groups of horsemen separately drawn up to move and fight on horseback as a unit. Instead, horsemen either fought at the side of their retinue, or (more commonly) served as mounted hoplites, riding to battle only to dismount and fight on foot. It took until the late 6th century BC for the first true cavalry to appear in Greece. They became ever more significant in Greek battles and campaigns; however, due to the small size of the Greek city-states and the high cost of maintaining horses, cavalry forces were of relatively limited size throughout the Greek world until the 4th century BC.

In the Classical period, Greek cavalry was essential in a number of roles: they guarded armies on the march, warded off enemy light troops, contained enemy ravagers and protected cropland, and covered retreats. Hoplites were extremely vulnerable to enemy cavalry due to the weight of their equipment and their lack of missiles. They lived in fear of mounted enemies, and relied on their own horsemen to protect them.

In battle, the Classical Greeks used cavalry in much the same way that Philip and Alexander did. They formed them up on the flanks of a phalanx or kept them in reserve, to cancel out enemy cavalry and to strike against any vulnerable point. Their equipment - heavy armour, javelins, lances and swords - allowed them both to fight at range and to charge home. Contrary to what you may have heard or read, cavalry action was frequently decisive in Classical Greek pitched battles. A quote like this -

No one knew quite how horses were used in warfare.

...is just so depressing :(

Here's some useful books:

• I. Spence, The Cavalry of Classical Greece (1993)

• L. Worley, Hippeis: The Cavalry of Ancient Greece (1994)

• R.E. Gaebel, Cavalry Operations in the Ancient Greek World (2002)

• P. Sidnell, Warhorse: Cavalry in Ancient Warfare (2006)