What do high and low air pressure systems mean in terms of density?

There are two physical differences between Low pressure systems and High pressure systems.

First, is the circulation surrounding them.

Secondly, is the atmospheric motion that they cause.

Low pressure systems circulate counter-clockwise. High pressure systems circulate clockwise. These "motions" are the building blocks in our atmosphere. They give us our weather.

A Low’s counter-clockwise circulation forces air upward (ultimately resulting in condensation, cloud formation and ultimately precipitation). A High’s clockwise circulation causes a sinking motion in the atmosphere, resulting in fair/clearer and often sunnier skies.