I promised I'd go more into teeth, so here it is.
Horses are grazers. They primarily eat grass and herbs. This gives them very different dentition from humans (who are omnivores).
Here's our horse skull again. We have a long, narrow jaw with six sets of molars, two sets of incisors, and one set of canines. No premolars.
A horse eats by cutting the top of the grass with its incisors and then using its tongue (which is quite large) to pull the grass up to those great big molars, which grind it into itty bitty pieces before swallowing.
The canines - which as you can see are small and not in opposition to each other - serve no purpose in feeding. In fact, about 25% of male horses and 75% of females don't have canines at all. Wild male horses might use their canines in fights, but they aren't even much use for that. Given more evolution, horses may lose their canine teeth altogether.
Premolars are a different matter. This horse, as you can see, has no premolars. Some horses do have one set of premolars - which we call "wolf" teeth (figures vary from 50% to 70%). In 90% of cases the wolf teeth exist only in the upper jaw. Many horses have "blind" wolf teeth - they never erupt through the surface of the jaw. Wolf teeth - especially blind ones - are often considered a problem. They can come into contact with the bit, causing unnecessary pain for the horse. Many horsemen automatically have the wolf teeth removed. Others elect to use bitless bridles with animals that have wolf teeth that are causing them problems. The wolf teeth are completely vestigial, and removing them has no long term effect on the horse.
Horses have open-rooted teeth that continue to grow throughout the animal's life. However, horses tend to outlive their teeth somewhere in their twenties and start losing them - it's possible to keep a "toothless old" horse healthy, but it requires special food, feeding soaked hay, etc. I've seen a horse stay in good weight with precisely one canine and about half a molar left in his mouth, so it's certainly possible.
Domestic horses may have their teeth grow faster than eating wears them down. We correct this by filing the teeth - called "floating" in the US and "rasping" in the UK.
Horses do not often get dental caries, but it does happen - especially if they're fed too many simple sugars. (Sugar lumps are fine occasionally, but bad in excess). Tooth abscesses and tooth loss can occur. In an animal that has lost a tooth, regular floating is even more important to prevent the opposite tooth from growing into the empty socket. (Bunny owners are familiar with the problem).