Air density is simply the amount of "air molecules" (actually a mixture of oxygen, nitrogen and a bunch of others) in a certain volume.
All these air molecules bounce around, bump into each other and bump into whatever happens to be in the air as well. This bumping into objects exerts a force on these objects which we call air pressure.
By their very nature, air molecules want to be as far from each other as possible, but there are a few factors that force them together. In the free atmosphere, gravity is the most important one: it sucks the air molecules to the surface of the earth. The air molecules resist this to some degree, depending on a number of factors. And it's these factors that eventually determine air density and air pressure:
- Temperature: the higher the temperature, the more "bouncier" the air molecules become. So they are able to resist the compression effect from gravity better, making sure that there are less air molecules in a given space. So when temperature goes up, air density goes down.
- Altitude: the higher up you go, the less air molecules are above you, sucked down by gravity and squeezing the air molecules below together. So the higher you go, the less dense the air becomes.
If it wasn't for the sun, the air density and air pressure on the planet would be even throughout the surface, and uniformly declining as you go up. But the sun heats the earths surface unevenly, causing spots where the air is less dense (due to a higher temperature) than elsewhere. Less air molecules means less gravitational pull so lower pressure. Nature doesn't like this pressure differential so air starts flowing from the lower temperature/higher pressure regions to the higher temperature/lower pressure region. Due to the coriolis force this air movement will not take a direct path but will "swing" to the right (northern hemisphere), creating the cyclonic and anti-cyclonic winds around high- and low pressure regions. And between the cold and warm air masses are the cold and warm fronts.
Now there are a few other effects, but on a broad level this is what causes "weather".