Air moves into the lungs when the air pressure in the lungs is lower than the air pressure outside of the lungs. This occurs when the muscels of the diaphragm contract.
You can see this by breathing in and out deeply. When you breath in you can feel your chest being pushed up and out by your diaphragm. The reason the pressure in the chest is lower is that when you breathe in you are increasing the volume of your chest. As there is the same amount of air in your chest the pressure decreases.
