Other types of mounts will vary, but horses have 4 movement speeds: walk, trot, canter and gallop. The speed given in their MM entry represents their speed while trotting; this is also used to calculate their overland movement speed. A walk is a slow pace, about as a man on foot, and is often used when trying to be especially cautious or when giving the horse a rest. A canter can be safely maintained for two hours, a gallop for one - the horse must be walked for an hour before its pace can be safely increased again.
Remember that these speeds are given in tens of yards, so multiply them by 30 to calculate how many feet are travelled.
In mounted combat, a character gets a +1 bonus to his chance to hit creatures smaller than his mount. Thus, a man on horseback gains a +1 bonus to his attack rolls against all medium-sized creatures such as other men, but would not gain this bonus against another rider or a giant. Those on foot who fight against a mounted rider, have a -1 penalty; this is not applied to attacks against the mount, however. Generally speaking, charging from horseback only changes how far you can charge as a result of your increased speed. However, if you make a successful Riding NWP check, you can charge an opponent, attack them, and use the rest of your horse's movement to carry on past them and out of their reach. If you fail, you are engaged and cannot escape. Only trained warhorses can be induced to charge an enemy.
Firing from the back to a still horse is a -1 penalty; walking or trotting is -2; galloping is -4. The Riding NWP reduces all these penalties by 1 per rank.
Lances are particularly dangerous from the back of a mount, and useless the rest of the time. As long as your horse is in motion, you can use the lance at the weapon speed and damage listed. Using a lance in combat requires you to ride at the enemy and strike them, then ride away and wheel round. As such, you can only attack every other round. However, you don't have to make a Riding NWP to disengage when using a lance.
The true power of the lance is unleashed when charging, causing them to deal double damage. Every time you use a lance in this way and strike a successful hit, however, it must make an item saving throw. Light lances are classed as thin wood, and save vs. crushing blow on a 13+. Heavy lances are thick wood, and save on a 10+.
Finally, a joust between two knights very commonly results in one or the other being unhorsed. If you charge another mounted opponent with a lance and hit, roll to see if the lance shatters. If it does not, roll a d8 and add 1 for every 5 points of damage dealt. If the result is above 7, the rider is unhorsed. If the target has the Riding NWP, they can make a check to reduce the knockdown die by 1 for every 3 points they pass the check by.
Untrained horses do not make any attempt to attack unless they are panicked. This kind of attack involves a lot of rearing and kicking, and will always dismount their rider. Warhorses will kick with their hooves on the second and succeeding rounds of melee. In this context, melee means that they are not moving. Refer to the table below to find the damage of various mounts; all horses have a weapon speed of 6 with their hooves.
With trained warhorses (or wild horses acting of their own volition), it is possible to trample a hapless enemy who is on foot. You need to be moving at a gallop to trample. The victim must make a flat saving throw vs. paralyzation to throw themselves aside. If they succeed, they are prone but safe. If they fail, they take maximum damage from both hooves. For horses that usually bite rather than kick, assume each hoof deals 1d4 damage.
Dangers of Mounted Combat
Panic & Untrained Mounts
When fighting from the back of an untrained creature, you receive a -2 penalty to hit. Furthermore, every time something startling happens - your mount is injured, a loud noise, etc. - you must make a Riding check or your mount panics. If you do not have the Riding NWP, untrained horses will panic 90% of the time when startled, and warhorses will panic only 10% of the time. Warhorses never panic for a trained rider. You might be thrown clear if a panicking mount is not saddled, but more often you will either be carried with it - or dragged by the stirrups. Mounts panic for 1d4 rounds, running away at 150% speed, and you can attempt to regain control each round with a Riding check. After the first round of bolting, you can also simply try to dismount, but unless you have Acrobatics this deals 1d6 damage, depending on how fast they are moving.
If you are unlucky, a panicking horse may unseat you and drag you around by the stirrups. This happens if you are unconscious, stunned, surprised, or under any other condition that would normally deprive of your reaction bonus from Dexterity. Every round, you take 1d4 points of damage; if you are wearing heavy metal armor, you only take 1 point of damage per round. If you have a bladed weapon of longsword size or smaller, you can cut the stirrups with an attack roll vs. AC 8. Otherwise, you can only free yourself with an Acrobatics check. If you have neither a bladed weapon nor the Acrobatics NWP, you can only try to yank yourself free with a Bend Bars/Lift Gates roll.
Riders can be dismounted if they are struck with significant impact. Any time a rider is hit by a weapon that is 3' or longer and the knockdown dice is successfully rolled, they are automatically unhorsed and take 1d6 damage unless they make an Acrobatics check. Being hit by crossbow bolts, percussive magic, low tree branches and so on may also require a Riding NWP check to stay on the saddle.