Okay I was looking into this and thought I would make a summery of it, feel free to agree, disagree, enlighten etc;
There were several types of horse that was in common use in the medieval period. And many had different purposes. I shall give a little into to each category and then explain more about them.
First of all then the most famous type the good old warhorse.
Chargers
Strictly speaking what we think of as a warhorse is generally the charger. A type that was often heavier and capable of carrying a fully armoured knight. The Charger was used for exactly what its name suggests, charging. Most made up of certain breeds, and usually stallions (a non-neutered male horse). They were typically split into three sub-categories
Rouncey's
The most common of the chargers, it was not as large or powerful as the more famous Destrier. However, it certainly had its place. It was generally more practical than the other charger types, often being chosen over Coursers or Destriers. Being lighter than the other two, it was naturally faster, and would often be used where speed was preferable, or where the ground was more broken. Probably the type used for 'medium cavalry' in the medieval 2. Most rich knights would own one of these as well as a heavier Courser. However, for the poorer knights (hedge knights) it was the only horse they would be able to afford. These were also sometimes used as packhorses.
Courser's
By far the most common of the heavy cavalry mounts. It was a very expensive horse. Larger than the Rouncey's and usually more spirited. They were not quite as fast, but were of a decent speed. They are what the vast majority of what knights would use. Especially on flatter ground, though would sometimes be substituted for Rouncey's in rough terrain. These are what you think of when you think of the knight’s barded warhorse. Able to carry not only a fully armoured knight but barding as well. Every knight worth his salt would own one of these.
Destrier's
The most famous of medieval mounts, this was a horse without peer for the battlefield. Very rare to be seen, due to its exceptional cost. One source places it at about seven times as expensive as a Courser. Its extreme strength and weight made it the most suitable of the chargers, and its speed being very nearly the match of a Courser's. To own one of these was a mark of status, as well as being damn dangerous on the battlefield. An estimate at its size places it at 15 or 16 hands.
Riding Horse's
No so much used in warfare, this is the type used by ladies, and gentlemen in their leisure pursuits. Trained in the trot and more rarely the amble (more later) they were probably the most comfitable of horses to ride. Used almost always in none war situations this was a horse most owned and rode. Can be either mares (females), stallions or gelded.
Hackney's
A breed that came about in 1300's England, was very common as a horse that was both powerful and versatile. Known for its great stamina it was able to hold at a trot for long periods of time. Generally became more popular after the medieval time period they nevertheless had their origins in this period. Typically replaces the Jennet in northern Europe.
Jennet’s
Originally a Spanish horse, it spread across Europe and eventually the America’s. A well mannered horse, with a comfitable amble, it was suitable for everything the more expensive Palfrey is. Used by the nobility as well as messengers or even merchants it was a popular horse.
Palfrey's
The most expensive of the medieval riding horses, it was something everyone who was conscious of their social standing sought to own. Trained in the amble, it was undoubtedly the most comfitable of horses to ride for any period of time. Used for a variety of things, from ceremonial duties, to riding distances, and hunting. It is sometimes considered to be equal to a Destrier in price, though the majority were less expensive than that.
Pack Horse's
Wide spread and the main type that was available to those not of the nobility. They are they half breeds, the Stumpter’s or the generic draft horse. These were used for work. Pulling carts for merchants, helping to plough the fields or just carrying goods.
Gaits
The speed at which a horse moves is typically divided into different categories. Based on its speed and the way in which the horse moves.
Walk
Fairly obvious, about the same speed as a humans walking pace, averaging about 4mph. A rider at this speed feels little discomfit but will experience a slight side to side swaying motion.
Trot
Most describable as a jogging speed for a horse, it averages at about 8mph. With a motion where the horse moves its legs in alternate diagonals (ie front right leg and the back left leg at the same time) it is a very stable speed for a horse. This is the speed at which a horse would travel for any length of time, most horses able to continue on for five or six hours, more if pressed thought that leads to the risk of injury. A reasonably comiftable ride for an experienced rider, there is nevertheless some discomfit if the rider is unfamiliar with this gait. Each time the horse places its legs, it sinks downwards slightly, this can cause a bouncing motion for the rider. Uncomfortable for the novice, but easy for a rider to avoid by straightening the legs slightly to avoid being bounced. This makes it harder on the rider for extended periods of time.
Canter
A decent speed for a horse, though one that it is unable to sustain for any substantial period of time. It is a pace where the horse moves in a three beat motion, pushing of with one of the rear legs, landing with another three, before bringing the original leg back into position. This is the gait where the horse appears to be leading itself with one leg, while the others move much more in time. The actual speed depends on the size of the horse and its leg length, but can be placed between 10 and 16mph. For a rider this is actually easier to ride. Since it doesn’t require the slight upwards push that the trot does, it does however require a swaying motion that takes practice to learn.
Gallop
The fastest speed for a horse, it can be anything up to 30mph. It is however a speed that like a person sprinting cannot be maintained for great periods of time, with most horses usually exhausted after a mile or so. However a slower paced gallop (named because of the motion of the horse rather than its speed necessarily) can be sustained longer. A movement that at first glance appears like the canter, it however has a forth beat, where the horse will place one of its feet slightly before the others. Riding this gait is difficult for a rider, since it combines not only the back and forth motion needed to ride a canter, but also the slight push of the legs required for the trot.
Amble
Most similar with the trot, the horse has been taught a new way of moving its legs, this results in a much smoother ride for the rider, though is slightly more tiring for the horse. Instead of the slight bounce caused by the placing of its legs, there is a slight pause between the placement of the legs, this creates the smoother ride.
The Medieval Horse
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The Maremmano (photo 
Noriker (photo Fausto Del Pin e Giorgia Ietri)
Male of Bardigiano (foto dr. Valentino Beretti - University of Parma)
