Pasties

Pasties (sing. 'pastie') are small pieces of cloth, plastic or metal. They are treated specially so they are sticky. They are used to cover a girl's nipples or genital area. They vary in size and are usually not much larger than the person's areola.
The concept of 'pasties' is to show as much of the breast as possible without making the woman entirely topless. Many exotic dancers in strip clubs are required to wear them by law. Pasties are usually applied with a special glue or tape. This is the origin of their name, as they are quite literally 'pasted' onto the nipple.
Pasties are worn by many neo-burlesque performers. Burlesque pasties may also feature tassels which hang from the center, performer will sometimes twirl these as part of a performance. 
Some women who choose not to wear a bra wear pasties to prevent their nipples from protruding visibly through their shirt. 
Pasties in law.
The United States Supreme Court held in "Erie v. Pap's A.M." that a city ordinance requiring erotic dancers to wear pasties, and a g-string, was constitutional.