A horse can have either bilateral cryptorchidism, where both testicles have failed to descend, or monorchid horses, with one testicle that developed correctly. In most cases, a bilaterally cryptorchid horse will have both testicles in the abdominal cavity.
Depending upon the exact location of the undescended testicle(s), a cryptorchid horse may still be able to successfully breed, although it is uncommon. Generally, if a horse has monorchidism, the properly developed testicle will have viable sperm, therefore allowing the stallion to breed. If the testicles are rendered ineffective due to location (and inability for the temperature to be regulated in order to allow for viable sperm) Testosterone will still be released, effectively allowing the behavior and temperament of a stallion, even though the horse may be unable to actually breed a mare.
It is still not fully understood what causes cryptorchidism, although a combination of genetic, hormonal, and mechanical factors are believed to all have some contribution.
It is possible to successfully castrate both a bilateral cryptorchid horse and a monorchid horse, although the surgery may be fairly risky depending upon the location of the under/undeveloped testicle(s).