There are a number of weather phenomena that cause damage from a major thunderstorm. Wind, rain and lightning are associated with almost any storm. The most damage, however, usually comes from hail and tornadoes in the strongest storms. Since hail and tornadoes are caused from similar effects they are often closely associated with one another.
The storms associated with these damaging events are caused by a strong vertical updraft. As heated air from the ground rises through the atmosphere the Coriolis effect twists it, counterclockwise in the northern hemisphere and clockwise in the southern.
As the warm air rises the pressure drops and, as anyone who has used canned air to clean their computer knows, the drop in pressure results in cooling. Combined with the warmer air meeting already cooled air high in the atmosphere the rising air cools and the moisture in the air condenses. This creates what is known as a cyclonic storm, a spinning mass of clouds rising tens of thousands of feet.
At times the spinning column of air can tighten. Like a spinning figure skater pulling in her arms, conservation of angular momentum causes the tightening air column to spin faster, resulting in a tornado. This fast spinning column of air can reach from high into the storm to the ground causing enormous damage.
Rain is usually associated with these storms as well. As the rising air cools, and the water vapor condenses into clouds, eventually the water droplets become large enough to form into raindrops. If the vertical shear or lift isn't strong enough to support these raindrops they will fall to the ground as rain.
In stronger storms, however, these raindrops are lifted higher into the atmosphere and freeze. As they freeze and rise through the clouds more water condenses onto the surface of the ice particles until they become heavy enough to again fall. Sometimes cycles of stronger updrafts will lift the ice particles again and again. On each trip up and down through the storm the ice particles accumulate more and more water, which is then frozen creating hailstones.
A tornado can create perfect conditions to cause very large hailstones. The updrafts associated with tornadoes are just what are needed to carry large pieces of ice high into the atmosphere. If these pieces of ice are carried away from the updrafts needed to create them and hold them in the air they will fall to the earth. This explains why hail often comes in advance of, and associated with, tornadoes.