Thunderstorms typically form in warm and moist air. To understand how they form, there are three basic rules you need to understand.
1.) The Sun heats the surface of the earth, which then heats up the air at the surface.
2.) Warm air rises
3.) The warmer the air, the more moisture it can hold. If warm air is cooled and can't hold the water vapor anymore, it will eventually condense (form a cloud) and if it gets even cooler, it will rain.
Let's start at the beginning of the day and imagine a cube of air. As the sun warms the surface our cube will become warmer than the air above it, and it will rise. Because of the cooler air aloft, our cube will start to cool (though it remains warm enough to keep rising). This cooling causes the cube to lose its ability to hold the moisture and a cloud forms and eventually rain forms. When this process happens in a violent manor, severe storms can form.
Rising air isn't the only thing that a storm needs to form. In order for a storm to become severe, it will need some shear. Shear is a changing in the wind direction and speed with height. This is not only important in forming a basic thunderstorm, but even more important in forming a tornado, which we'll cover later.
Hail of more than 1" in diameter is considered severe. Hail forms within a storm because of the updraft that causes the storm (the rising cube of air). As air moves upward, rain drops that are falling are pushed back up into the cloud. At some point, the rain crosses above the freezing line (where the air is below freezing) and becomes a small chunk of ice. The small chunk of ice then falls again, picks up a few raindrops and the updraft pushes it above the freezing line. As the new rain drops freeze to the original chunk of ice, it gets bigger. This process is repeated until the chunk of ice is too heavy for the updraft to support or the updraft falls apart. The chunk of ice then falls to the earth's surface and is called hail.
Wind created from a thunderstorm of 58 mph or more is also considered severe. There are a few ways that wind is created with a thunderstorm, but the most common is just the opposite of what creates the thunderstorm, cool air. As the storm rains heavily, the air aloft goes through evaporative cooling, meaning that as the storm rains heavily, cold air will form aloft. When the updraft is strong enough that cold air will quickly rush to the ground and spread out. This is known as a downburst.
Another way severe winds can be created is through mixing. Imagine a scenario where the jet stream (typically with a wind speed of at least 70 mph) is close to the earth's surface. As warm air rises, the wind aloft then is "mixed" down to the surface of the earth. Because of the compression of the jet stream (higher pressure at the surface) the jet can actually accelerate.
The other is represented by the graphic to the left. Imagine a rotating tube of air caused by shear. As the storm starts and the updraft forms, the rotating tube is turned sideways. That rotating tube then can start the process of rotating towards the surface of the earth and create a tornado.
The process that creates a hurricane is similar to thunderstorms and tornadoes, but not completely the same. For us here in the U.S., we're typically worried about hurricanes that form in the Atlantic, known as the Atlantic Basin. Because the Atlantic Basin is actually further south, storms move from east to west, rather than the east to west we typically see in central Illinois.
As a disturbance, such as a cold front aloft (above the surface) or a wave coming off of the African coast moves over warm water, a hurricane can form. This is done by the warm, moist air rising into the disturbance and rotating counter-clockwise. As long as the disturbance can move over more warm water, it will likely continue to strengthen. Unlike thunderstorms, hurricanes don't want shear, because they can tear the disturbance that creates them apart.
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