When people use spears to fight cavalry do infantry stab the horse or the rider?

The horse always the horse. The reason Calvary decimated infantry and especially archers was a war horse could weigh 1,200 and 1,400 pounds and can run at speeds up to 55 miles an hour. A rider on such an animal could ride right over massed foot soldiers killing or incapacitating many men, not by act of the rider but just the impact of the horse. You armor that horse and it becomes even more effective killing machine.

War horses were highly trained to charge and wade into foot soldiers. That's why good war horses were super valuable. In the 11th century swathes of land and titles were being traded for horses.

Medieval Warfare 1000-1300
Between 1020 and 1030 the Abbey of Jumieges paid Hugh Bishop of Bayeux a horse 'of great price' for land and privileges at Rouvray. In 1030 it paid Drogo Count of Amiens and the Vexin six horses 'of very great price' for land at Genainville. In 1045-1048 it paid Roger de Montgomery a horse worth 30 pounds, together with a cuirass worth seven, for the land of one of his vassals. About 1054 Gilbert Crispin was paid 200 pounds in money, 2 ounces of gold and a horse woth 20 pounds for the fief (beneficium) of Hauville on the edge of the forest of Brotonne.

People paid their taxes with them. And most war horses were smaller and faster not the larger heavy horses which would follow. Heavy Horses associate with knights were a late development. The Normans, who took knight warfare to Britain for instance, rode horses similar to Arabian mares(small). Around 15 hands.

As jFrankcar said in his fine answer. The reason to use a spear against a charging horse wasn't to throw the spear and hope for the best, but to stand on the end of the spear, planting it into the ground, along with others in a formation and present a wall of spears to any hors*m*n hoping to run you over. The horse would be impaled on hopefully several spears and die, and you would get to live. If a pike man impaled the rider the charging horse would surely still kill him. In order to impale the rider a pikeman would have to be almost right in front of the charging horse. If the pikeman got the rider and he would never know it because the horse will have killed him probable in the same instance he killed the rider.

The objective was to present a wall of spears to protect your foot soldiers from mounted cavalry charges. Typically archers because they were more valuable than infantry. Also because Archers were required to be a bit more nimble to work their bows than were infantry and thus they were less heavily armored and more susceptible to cavalry charges.

from Pieter Geerkens That top speed is for a quarter horse - strictly a 19th century bred in America breed. Top speed for a thoroughbred carrying just 130 lbs or so (jockey and saddle) over 2 furlongs is barely 45 mph. When you get to less specialized and larger breeds carrying a more typical rider and kit the speed is closer to 30 mph or 50 kph. –

Well the Kentucky derby is 10 Furlongs 1.25 miles and while The typical thoroughbred horse is capable of running only a quarter of a mile (400 meters) or so at its top speed. Horses at the Kentucky derby do run 45 mph for the entire 1.25 miles. which means 45 mph is not their top speed.

the thoroughbred is the fastest breed of horse in the world, and can maintain a speed of 45 miles (72 km) per hour for a distance of more than a mile (1.6 km), making the Derby's 1¼ mile-long race the fastest two minutes in sport

Pieter the sourced speed is for [Equus ferus caballus](Equus ferus caballus) which is not a Quarter horse exclusively. Perhaps you are speaking of a sustained speed over a mile or more of 45 mph (triple crown races are all over a mile).. My SOURCE and my intent described a top speed, not a sustained speed.

Spears are (relatively) long ranged weapons to use against cavalry. Bows and arrows are even better (as at Crecy and Agincourt in France), but only if they can be fired at a high rate, and at long range. Most infantrymen were not skilled enough to do this, which is why they used spears. They could try to kill the horse or the rider, but usually the horse was the bigger, more vulnerable target. Kill the horse, and the rider comes crashing down, at your mercy.

The cavalry had the advantage of speed. In some cases, they could gallop behind the defending lines faster than the infantry could turn around, and take the infantry in the rear. That's why terrain was so important. Flat, easy, terrain favors the cavalry. Hilly, forested, or otherwise broken ground slows the cavalry charges and helps the infantry. At Agincourt, there was a rainstorm before the battle that turned the ground into mud. It slowed both British infantry and French cavalry, but the cavalry suffered RELATIVELY more.

A group of men armed only with spears were usually not enough to fight cavalry by themselves. The successful ones usually had something else going for them, such as armor and/or shields. Sometimes, they would "mine" ground against the cavalry, by digging "potholes" or setting up sharpened stakes in front of the infantry lines.

Edit: In response to the commenter, cavalry would almost always win under ideal conditions, flat ground, good weather, few obstacles etc., that allowed them to make good contact with the infantry using 90%+ of their men in a simultaneous assault advancing at double digit mph against poorly deployed (e.g. outflanked) infantry. They would lose if broken ground, weather, or other obstacles led to poor contact using a fraction of their hors*m*n in piecemeal attacks with a speed in the low single digits against a square or other fully formed infantry formation. As the commenter also pointed out, horses were less keen on running a suicidal charge than men, so deploying the spearmen to threaten the horses, (e.g. lower aim), was the better idea.