One of the major ingredients needed is something called wind shear. Wind shear is the change in wind direction and speed from the surface up to couple thousand feet in the atmosphere. Picture a horizontal tube of wind at the surface. The wind at the surface is slower than the wind aloft due to the friction at the surface. This causes a horizontal wind tube to form. The wind shear helps tp start the rotation that is needed for a tornado to form.
Another important ingredient needed is an updraft (rising air). This is where a thunderstorm comes into play. A mature thunderstorm has both an updraft and a downdraft. The updraft in the thunderstorm forces the horizontal spinning tube at the surface to rise and become vertical. The verticle spinning tube continues to spin and allows a funnel cloud to form. The funnel cloud then continues to grow until it reaches the ground and becomes a tornado.
The faster the winds are aloft the faster the vertical tube will spin and the more powerful the tornado will be.