Both, but head-on wasn't the ideal way of delivering a cavalry charge. The ideal way of delivering a charge was to the flanks or rear of an enemy unit. While infantry were engaged head to head, the cavalry would find an exposed flank where possible and deliver its charge there. This would minimize the casualties for the cavalry and multiply the casualties, confusion and panic to the receiving unit of the charge.

Reason being, at the flanks and rear of a formation, there would be little defense against an attack. Especially if that unit/formation was already engaged in a battle, its attention being diverted to whatever threat it's already engaged in. The unit would be unable to break-off its current engagement to meet the new threat, especially against a fast, mobile cavalry formation.

Now depending on the unit on the receiving end of the charge, if it was disciplined, well led and trained, it could withstand the charge and defend against it. On the other hand, a recently raised/conscripted unit with little training would possible break before contact. A mass of galloping horses is a frighting site, for a recently conscripted peasant it would be enough to convince him to run. At the same time, a well trained, disciplined unit engaged for hours in close combat during a battle, suffering casualties and bordering on exhaustion would probably not fair well against a well timed and properly delivered charge (flanks, rear of formation).

A charge delivered to the rear or flanks of a formation would create chaos, confusion and panic. The men at the front or opposite side would be powerless to do anything about it. Those engaged in combat at the front with the original threat would now be faced with two threats. Knowing their security is now compromised and the enemy is at their backs, they would try to seek safety, i.e most likely run for it, or try. Due to the nature of combat, especially prior to the modern era, battlefield awareness/information was extremely low. Much more so for the individual soldiers or even a unit/formation. So, when a unit was on the receiving end of a cavalry charge, especially from the rear. They could automatically assume that their neighboring units have been routed, defeated and fled. Which could also mean that the battle is not going well. So in turn, the soldiers would see all hope is lost, and flee.

Through out history, cavalry were used and implemented on the battlefield in different ways, depending on the situation and the general leading them. Sometimes a cavalry unit was hidden in an ambush position prior to battle (out of site of the enemy) so it could charge at the right moment when the infantry was engaged (Battle of Trebia, Mago's ambush). Other times the commander would hold his cavalry in reserve, waiting for the right moment. Possibly waiting for his opponent to make the first move.

Sometimes cavalry would feint a charge and withdraw, the infantry thinking the cavalry is retreating would pursue. Only for the cavalry unit to make a quick 180 and charge into the now looser or broken up formation of infantry.

And yes, at times cavalry did charge head-on into an infantry formation. Of course as you can imagine the results varied between highly successful to disastrous. A dense spear formation would easily repulse a cavalry charge as the horses would be impaled before even reaching the actual soldiers. A loose broken up formation, or lightly armed with minimal armor would not fair too well.

I am not a professional historian, but I've read:

The Fall of Carthage - Adrian Goldsworthy

1914-1918 - David Stevenson

How Rome Fell - Adrian Goldsworthy

The Renaissance at War - Thomas Arnold

The Punic Wars - Nigel Bagnall

Edit: added a bit more of an explanation on battlefield awareness and fixed spelling.

Cavalry was less used as a "shock" direct assault weapon and more as a tool of exploiting an enemy's weakness in the majority, whether it be in battle or outside. Military technology, namely the stirrup, was also not extant at this point to allow heavily armed and armoured soldiers to devastate an enemy line in the way that they would become famous for the Medieval age. Classical sources are notoriously vague on cavalry but some sources give us insight into this debate.

With Greek cavalry, the historian Thucydides suggested in the fifth century BC that cavalry had a dual-role. One was as a "mobile defence" i.e. getting soldiers to other areas quickly during battle. The second was to harass a foraging army. Supply trains were not extant in the classical period and cavalry therefore took advantage of the fact that the army had to resupply itself by leaving the safety of camp.

Concerning Roman cavalry, Vegetius wrote De Re Militari in the late fourth century AD, drawing on a now lost work by Cato from the early second century BC. Vegetius does distinguish between two types of cavalry: heavy and light, but concentrates much more on the latter. The key role of light cavalry was to threaten and harass enemy wings in battle. Vegetius saw the role of heavy cavalry as a protector of infantry flanks, rather than taking the fight to the enemy with a direct assault.

Parthian cavalry are talked about with respect to their role in the crushing defeat of a Roman army at Carrhae in 53BC. While armoured Parthian cataphracts were effective in directly attacking the Roman infantry, this force was secondary to the mounted archers that sought to cause disorder. Both of these forces contributed to the crushing of the Romans at Carrhae but were not the sole cause. Cavalry alone could not cause a military defeat, only create and exploit a weakness to facilitate it.

The Greek, Roman and Parthian cavalry use all present a similar argument, that cavalry exploited an enemy's weakness and vulnerability and were much less used as an assault force. The third century scholar, Aelian, sums up well in seeing wealth of cavalry in exploiting a weakness in the line, and refers little to any idea of a heavy cavalry actually turning the tide of a battle. Prior to the invention of the stirrup, cavalrymen were unable to wear heavy armour without likelihood of falling off the horse from the weight. For the same reason, lances could not be used to the devastating effect they would later become known for in the middle ages. Direct assault by a shock heavy cavalry force in order to change the tide of a battle was not as common as games such as Rome: Total War would suggest.