The correct name for it is rubatta, and since Punjabis have some problem in pronouncing Urdu words correctly they call it dupatta. If you were to listen to old hindi film songs you will hear the word rubatta, and not dupatta.
It is traditionally worn across both shoulders. However, the dupatta can also be worn like a cape around the entire torso. The material for the dupatta varies according to the suit: cotton, georgette, silk, chiffon, and more. The other names for dupatta are orni, chunri, chunni and orna [mainly in Bangladesh](sometimes shortened to 'unni' by many Gujaratis).
There are various modes of wearing the unsewn dupatta. When not draped over the head in the traditional style, it is usually worn with the middle portion of the dupatta resting on the chest like a garland with both ends thrown over each respective shoulder. When the dupatta resting is worn along with the salwar-kameez it is casually allowed to flow down the front and back.
The use of the dupatta has definitely undergone a metamorphosis over time. In current fashions, the dupatta is frequently draped over one shoulder, and even over just the arms. Another recent trend is the short dupatta often seen with kurtas and Indo-Western clothing. Essentially, the dupatta is often treated as an accessory in current urban fashion. Nevertheless, the dupatta remains an integral part of Bangladeshi, Indian and Pakistani clothing.