For heavy cavalry, the purpose was to basically stampede or trample foot soldiers, at least in the intial part of the charge. You have an animal that was, conservatively, a thousand plus pounds fitted with armor mounted by a man that was armored as well and holding a large pole arm. There was a tremendous amount of weight in motion there and a single man or even a couple were mostly helpless to stop it, though with the correct weapon (ideally, a spear of sorts) they could still inflicted great damge. This is one reason commanders avoided charging cavalry head long into a spear formation, preferring rather to take the flanks or even better, the rear. What were the benefits of cavalry charges? Firstly, shock. The sound of numerous heavy war horses pounding the ground and charging at you, we could only imagine, would be very unnerving. As well, getting hit or seeing another get hit by this enormous animal or the lance of it's rider as it was charging full speed must have been gruesome and frightening to behold. Secondly, horses charging through the lines caused disorder in the rank and file and made them more susceptible to any number of bad circumstances that come from poor order on the battlefield. But cavalry wasn't all charging. Heavy cavalry would sometimes get "stuck in" when their charge didn't penetrate or cause disarray as planned or commanders would purposefully send them in for prolonged engagment. When this happened, they operated much like foot soldiers in that they held a tight formation and eliminated as much space between them as possible. This helped to prevent the enemy from separating and surrounding individuals, in which case they would become easy prey.

Your first part I reserved for the last part of my comment, because the Mongolians utilized mounted troops a bit differently than a lot of other major powers, or at least European ones. Though the Mongols did use heavy cavalry, they employed mounted archers extensively and were reknown for the training and application of them. Obviously, they would be different than heavy cavalry for many reason. Firstly, they used smaller faster horses to maintain a range between them and (hopefully) even the fastest of their enemy's cavalry units. These horse archers were great a shooting on the move and with greater flexibility than most other horse archers. Firstly, because they used a thumb-release technique on their bow. It's hard to describe without seeing it, but by pulling the string with their thumb, they could use the other fingers to put pressure on the arrow and keep it on the bow when shooting in a variety of odd angles. Here, like this. With the Mediterranean draw, the arrow will fall off the string if you tilt bow the right way. As well, thumb technique decreases the likelihood of fouling the release if you don't let go with all three fingers at exactly the same time. I won't elaborate on all the things that made them great archers, but you get the point. With all of these skills going in the favor, they became great at feigned retreats, harassing and demoralizing troops, enticing impetuous commanders to commit to less than favorable courses of action, etc.

I'll touch on light cavalry briefly. Light cavalry were predominently used to harry skirmish troops or lightly armored ranged units, run down retreating units, reconnaisance, skirmishing, or occasionally, charging wavering units. Obviously, they were outfitted lighter than heavy cavalry and with some different kinds of weaponry than would be common for their heavier counterpart - javelins come to mind. These were troops you probably didn't want in an kind of prolonged combat.