There are actually three types of thunderstorms, each producing different levels of impact: Single-cell thunderstorms, multi-cell thunderstorms, and supercell thunderstorms.
First off, the process of hot air rising and falling air is called a convection cell. Convection cells are when there is a balanced upward rise of heat and a downward fall of cooling air.
The most common, what you would refer to as an ordinary thunderstorm, is a single-cell thunderstorm. These have just one convection cell, and are typically short-lived and small in size. They can bring heavy rain and lightning with them.
If multiple cells move in a cluster, these can form a thunderstorm - a multi-cell thunderstorm. These form along a cold front, where hot air is up above and cold air is near the surface. These can bring drenching downpours and strong wind gusts. If conditions are right, they can form a squall line, which is an organized line of thunderstorms that stretches for as much as 600 miles.
The third category is called supercell thunderstorms. Supercells happen when there are deep updrafts within the thunderstorm. These can last for hours and drop torrential rains, often with hail. In these storms, air sometimes moves as much as 175 mph upwards, creating a sharp rotation. These are the storms that produce tornadoes, but thankfully, this type of thunderstorm is the least common.