Horses only have incisors, molars, and sometimes canines (the last more common in males). They have long since lost their premolars.
This is what creates the gap we put the bit in.
However, about 70 percent of horses have vestigial premolars - we call these "wolf teeth". Why? Because traditionally wolves are bad.
Wolf teeth generally erupt between 5 and 12 months of age and they're positioned right in front of the first molar in the upper jaw. Wolf teeth in the lower jaw, however, tend to come into contact with the bit, often causing discomfort for the horse. Blind wolf teeth, that have not erupted or only erupted partially, are usually further forward and cause even more problems.
Because of this, wolf teeth are generally removed when the animal is a yearling, before ridden training starts. Blind wolf teeth, however, may not be recognized or identified until the animal is trained to a bit - and they tend to cause the worst problems.
In modern times, wolf teeth are extracted with local anesthetic and standing/conscious sedation, and by cutting the gum and removing the tooth with forceps - it's slightly more complicated than tooth extraction in humans because the tooth is so vestigial. In colts, it's not uncommon to remove wolf teeth at the same time as castration.
In older times, they were removed by banging at the root of the tooth with a hammer until it falls out - a method now considered cruel, but if you're writing a more historical piece it might be the only way to do it.
If the horse is already in training, they generally need seven to ten days of not being worked in a bit so their gums can heal. (Some people use a bitless bridle rather than resting the horse for that long).