pV = nRT
That's it! Let me explain, and then the attached actual weather chart from 5pm CST this evening will make all the sense in the world.
To not put you through the torture of deriving the equation, it can be simplified as simply saying:
Air Density = Air Pressure ÷ Air Temperature
All this states is that Temperature effects Pressure and Density. If the temperature changes, the pressure changes and the density (the weight of the atmosphere) changes. Likewise, if the pressure changes, so must the temperature and the density.
The first thing we all learned in science back in elementary school is that warm air rises and cold air sinks. If there is cold air at the surface (like in Wyoming and Idaho right now), then the air in the atmosphere sinks. As it sinks, more air fills in the gap. This causes more molecules to fill in, thus making the atmosphere "heavier" (i.e. more dense). The heavier (or more dense) the atmosphere, the more pressure at the earth's surface. That's why there's always a big blue "H" on the surface weather chart somewhere near the coldest place that day.
Since the earth is constantly trying to fall into equilibrium (just like when you pour hot water into cold water and they make luke-warm water), these pressure differences cause what is known as the "Pressure Gradient Force". Now this is a different physics equation, but all you really need to know is that it causes the wind to blow. ...which in return starts moving all the other pieces of weather phenomena.
Weather = Sun Heats Earth Unevenly= Pressure Changes Unevenly = Uneven Pressures Try To Equalize