This is another big one. I'm thinking a lot about feet right now because I can't find anyone to take care of my horse's and she is now way overdue. That will all make sense by the end of this post.
To start off on the most basic level, the foot of the horse is called a hoof. It's a keratinous structure, like fingernails and hair, surrounding a bone. There is a membrane inside that acts like a sponge, pumping blood through the hoof. At the bottom of the hoof is a V-shaped piece of soft tissue (soft only in that it's not bone or hoof) called the frog. The frog is made of what feels like hard calloused skin, but can still be easily cut or damaged.
There are channels that lie on either side of the frog and these must be cleaned out with a hoof pick regularly to ensure that no rocks or other debris is stuck in there. A rock stuck in the hoof can cause sole bruising, which can be very painful and make the horse lame. If the frog is cut too deep, it will bleed and the horse will limp. The horse also must be cared for to ensure that there is no chance of infection.
For your reference, here is a picture of the sole of a horse's foot:
The horse pictured above is unshod, meaning, it has no horseshoes on.
Horseshoes are used for horses who travel long distances on trail or on concrete or asphalt, to give the horse extra grip and prevent their hoof from wearing down too quickly. They are used for corrective shoeing, which means that they help correct hoof shape or angle or one of the many other myriad things that can make a horse go lame. The most common type of horseshoe rims the edge of the hoof with the exception of the frog, and is nailed into the hoof wall, so while the hammering can cause some discomfort, it's not painful for the horse to be shod. There are other special types used for corrective shoeing, but I won't go into those here.
I will also say that horseshoeing is a highly contested area of horse ownership. This is my knowledge, and I'm trying to make it as general as possible, but it is quite possible you will hear other opinions. This area tends to fall more under what you believe than anything else, though I personally am of the school of trying whatever you need to until something works.
The hoof is the most important part of the horse. In the wild, horses run or walk about constantly. Their hooves grow and are worn down by the consistent movement and rough terrain. Domestic horses don't necessarily move that much and must have their feet trimmed every six to eight weeks by a professional, called a farrier or a shoer. Most farriers are also blacksmiths-- they have portable forges and will shape and heat shoes on site for the particular horse they're working on.
But if one foot is hurt, injured, or tender, the horse will be crippled. Most horses weigh upwards of 1,000 lbs. Imagine that much force even on four feet, and carrying that much weight on an injured leg equals some serious pain. Since horses are herding, prey animals, if they cannot run away from a predator, they are as good as dead.
You might think this isn't true in domestication, but horses were built to stand-- they only lie down briefly to rest or sun, or for longer periods of time if something is wrong (a horse lying down is often the first sign of illness, injury, or impending birth). Their rib cage isn't built to support long periods of time lying down, and can have compromising effects on their digestion and respiration. So a lame horse, even a domesticated one, is in serious trouble. Especially if they go lame far from home.
Lameness from the hoof can be caused by any number of factors: the above mentioned rock in the sole, diet, bad hoof care, and so on. There are also any number of other reasons a horse might go lame unrelated to the hoof: stretching or injuring a tendon or muscle, etc.
A lame horse should generally not be moved more than necessary if it has a hoof injury. And they definitely should not be ridden, or asked to do work of any kind. These can greatly aggravate an injury and turn it from something minor to something major rather quickly. Not to mention, it's just mean.
So there you have it. Hopefully you understand now why I'm upset that I can't get a farrier to do my horse's feet! If you have any particular questions or want something cleared up, leave it in the comments!