Does the appearance of a tornado come from the dust or is there some characteristic of swirling wind? Additionally many experiments shows a tornado-like phenomena in a jar of water, is this possible without water?
The characteristic funnel of a tornado is caused by condensed water. A tornado is a swirling mass of air with very high vertical vorticity and a corresponding drop in pressure. Large tornados can have pressure deficits on the order of 100 hPa, which is significant compared to synoptic horizontal pressure gradients. The drop in pressure causes air to expand and cool and if the air is humid enough this will be accompanied by condensation and the visible appearance of the funnel.
It is also worth mentioning that the visible tornado funnel does not have be in contact with the surface to be a tornado, it is only required that the surface wind circulation is connected. In these cases the lowest part of the tornado may remain invisible.
As a tornado persists the funnel cloud can pick up debris (dust, dirt, plants, parts of destroyed structures, etc) and some of this debris can orbit the tornado (other parts may end up lofted into the upper troposphere or centrifuged out at low levels) adding to its visual presentation.
The shape of a tornado is due to the movement of air and not dust. The shape of a tornado is called a vortex. A vortex can occur in any fluid (air or liquid) where the fluid rotates about an axis line. Vortices are examples of turbulent fluid flow (air or liquid).
When vortices occur in liquids they are sometimes called whirlpools. Tornados are high energy air vortices, less energetic air vortices can occur as dust devils.