So, why do we take a piece of metal and nail it to the bottom of a horse's hoof? Part of the answer is - we don't always. Many working horses, especially ponies, do their jobs fine without shoes. And some modern horsemen prefer to put hoof boots on only for work and leave horses unshod the rest of the time.
The actual answer? There are three reasons we put a shoe on a horse's hoof:
1. To prevent the added wear and tear of work - from the weight of a rider or hard surfaces - from wearing the horse's hoof faster than it can grow. This is why many ponies and mustangs aren't shod. If a horse's hoof can handle the wear without shoes it's generally better (and cheaper!) not to shoe.
2. To give a horse extra traction when working. Specialist shoes are designed for different disciplines and some horses are shod with studs - these screw into the hoof or are permanently attached and stick out a little bit. Carriage horses are shod with road studs designed to improve grip on tarmac. Jumpers have studs put on their front hooves to help them "dig in" after a jump and basically stick their landing. Racehorses are also shod with studs to help them get better traction at speed.
3. To correct hoof and foot problems. Special shoeing (called surgical shoeing) can be used to help ease the pain of horses with certain conditions, most especially navicular syndrome (which deserves its own post) and also to correct issues with a horse's action. Club foot can also be partially corrected by trimming and shoeing when the horse is still a foal.
But, again, many working horses never have a shoe put on their feet. I'll talk about how we can nail a shoe to a horse's foot without bothering it tomorrow.
Draft horses like this Belgian often have special shoes designed to increase traction when pulling.