Leotard materials
Most leotards are made of velours of lycra fabrics.
These fabrics are used because of their ability to stretch; this is usually a four-way stretch. The stretching is necessary because in gymnastics and dancing your body bends in ways which would cause normal or less-than four-way stretch materials to tear; and thus, become ugly. You can imagine that quality and proper materials are extremely important for leotards.
Lycra
Lycra is a smooth and shining fabric, which you will mostly see in competitions and important events, because it adds a shine (bling bling) factor to the leotard.
Velours
Velours is a softer fabric, which feels nicer to the touch. Velours leotards are mostly used for training or recreational purposes.