What is the relationship between air pressure and air density?
Air pressure, or atmospheric pressure, is defined as the weight of air in Earth's (or another planet's) atmosphere. Pressure is a force exerted on or against an object by being in contact with a substance. Thus, air pressure is due to the bombardment of the air (gas) particles with a surface.
Density is defined as the mass per volume of a substance. Thus, air density is defined as the mass of air per unit volume.
Using these definitions, we can answer your question. The more gas particles there are within a given area, the greater the air density. Likewise, the greater the air density, the more gas particles there are per unit volume. Therefore, there are more air particles bombarding with a surface. Thus, the air pressure increases as the air density increases and visa versa.
Using this logic, one can conclude that there is a direct relationship between air pressure and air density.