There are three types of ballet shoes: soft ballet slippers; pointe shoes with hard toe boxes and strong shanks; and demi-pointe shoes with toe boxes and softer material elsewhere. Pointe shoes, also called toe shoes, aren't pre-shaped for the left or right foot. They're immediately recognizable from the square toe box at the tip, and each shoe is entirely covered in fabric, topped off with ribbons on each side that allow dancers to securely tie it to the foot.
History
Dancers have been performing en pointe, or on their toes, since 18th century French choreographer Charles-Louis Didelot first incorporated the style in his works. More an acrobatic element than a fluid aspect of the overall ballet then, pointe work came into its own in 1832, when Italian/Swedish ballerina Marie Taglioni performed on toe for much of La Sylphide. However, famed Russian ballerina Anna Pavlova helped pioneer the construction of modern pointe shoes by customizing her own with additional durable material in the toe area to give her atypical feet better support.
Function
Pointe shoes are designed to permit dancers, primarily female, to perform ballet steps while standing on the tips of their toes. To achieve that feat, they're constructed with several features not found in ordinary soft ballet slippers. Yet, the main purpose of the shoes is not specifically dancing on one's toe tips, but rather making wearers seem as weightless, and therefore as graceful, as possible.
Toe Box
The most significant and distinctive part of a pointe shoe is the toe box that surrounds and supports the toes. It has evolved over the centuries from near nonexistence--Taglioni wore plain ballet slippers with minimal stitching reinforcement during La Sylphide--to a hard and durable enclosure with a flat front end that permits complex and prolonged steps as the dancer balances on it securely. Toe boxes are made with numerous firm materials, such as leather.
Shank
Next in importance is the shank attached to the soft sole on the bottom of the pointe shoe. Constructed of a sturdy material, such as plastic or leather, the shank provides crucial support for the arch in a dancer's foot. The material used also often allows the shoe to come in varying degrees of rigidity, such as extra strong or flexible, to permit different styles of toe work.
Demi-Pointe Shoes
Demi-pointe shoes are a hybrid ballet shoe worn in very specific circumstances. They have no shank, and while they do include a toe box, it is both softer and shallower than that found in pointe shoes. Due to the reduced support in the arch and toe area, dancers use demi-pointe shoes when only the appearance of a pointe shoe is necessary, such as ballet roles where pointe work isn't actually performed, and when familiarization with how pointe shoes feel is beneficial, such as before pointe training begins.