So. We shoe horses.
Which means that we nail strips of iron to the bottom of their hooves. Often strips of heated iron. The layman looks at this process and goes "How come that doesn't seem to bother the horse?"
The answer is quite simple. A horse's hoof wall is made of keratin - the same stuff as hair and fingernails. When we trim the hoof - which we also do with domestic horses in order to get what we call "good angles," that is to say a hoof shaped correctly to stand up to work and to prevent the hoof from growing faster than it's worn down - it's no different from trimming and filing your own fingernails and, indeed, uses similar tools. Larger, but similar.
When we nail on the shoe, the nail is driven entirely into the outer hoof wall and through it, then tapped down to secure the shoe in place. Unless the farrier screws up, the nail goes through only dead tissue. The horse probably feels something - tap on one nail with the other and you might get a similar sensation - but they rapidly get used to it.
Now, sometimes the farrier messes up and the nail hits the laminae - the tissue supporting the hoof. We call this "quicking" a horse. It's very rare for this to happen - a trained farrier knows what he (or she) is doing and the horse will very quickly let you know if you're putting any pressure on. Usually, the farrier will then withdraw the nail and place it correctly. A quicked or close nail left in can cause a hoof abscess. So if you want to cause a bit of drama...and a lame horse.
Quicking can also sometimes be caused by a poor quality hooves or by the horse being an idiot while being shod. But it's still very rare and, if anything, is rarer in lower tech societies where really competent farriers are more common.
A note for people setting their stories at the race track: Racing "plates" as they are called are made of aluminum or even plastic to make them lighter. Many trainers also reduce the weight further by getting rid of the nails and glueing the shoes on. Some riding horse owners have also moved to glued on shoes because they feel that the hooves are healthier without all those holes in them.