From National Weather Service:
Most tornadoes are spawned from supercell thunderstorms. Supercell thunderstorms are characterized by a persistent rotating updraft and form in environments of strong vertical wind shear. Wind shear is the change in wind speed and/or direction with height. The updraft lifts the rotating column of air created by the speed shear. This provides two different rotations to the supercell; cyclonic or counter clockwise rotation and an anti-cyclonic of clockwise rotation. The directional shear amplifies the cyclonic rotation and diminishes the anti-cyclonic rotation. All that remains is the cyclonic rotation called a mesocyclone. By definition a supercell is a rotating thunderstorm. When viewed from the top, the counter-clockwise rotation of the mesocyclone gives the supercell its classic "hook" appearence when seen by radar. As the air rises in the storm, it becomes stretched and more narrow with time. The exact processes for the formation of a funnel are not known yet. Recent theories suggest that once a mesocyclone is underway, tornado development is related to the temperature differences across the edge of downdraft air wrapping around the mesocyclone. However, mathematical modelling studies of tornado formation also indicate that it can happen without such temperature patterns; and in fact, very little temperature variation was observed near some of the most destructive tornadoes in history on May 3, 1999 in Oklahoma.