The hoof wall is made up of a protein called keratin which forms the horn tubules which have some spring to allow 'give.' An area called the bars prevent the hoof from spreading too much. It bears the majority of the horse's weight. It grows and wears away continuously.
The sole is slightly softer than the hoof wall. It should be concave because the less weight it bears the greater the traction. When it is concave the hoof can almost grasp the ground. It grows to a certain point then sheds.
The frog is triangular and even softer than the sole, almost elastic. It should only bear weight during movement. This assists blood flow, traction, and absorbs shock. It sheds off like the sole.
The white line marks the connection of the hoof wall to the sole and helps guide a farrier as to where to place the horse shoe nails.
Some nutrients associated with hoof growth are vitamin A and Biotin. Vitamin A helps hold horn tubules together. Biotin is supposed to improve the quality of the hoof wall. However this is dependant on the amount administered, a sufficient amount of zinc and methionine (horses can't use biotin without high enough levels of these minerals), and long term use. It takes about a year to grow a new hoof. The effects of biotin supplements cannot be seen immediately.
There is a small controversy about whether humans should trim the sole and frog. If the terrain is rough old tissue will be worn away naturally. On the other hand, if the horse stands in a stall most of the day, day after day there is nothing to wear away the old tissue. If the horse doesn't seem to be able to shed it by himself then human aid may help. If the horse tends to have a thin sole or gets sole bruises a lot then trimming may do more harm than good.