The compliance of the lungs and chest wall is determined by how elastic they are. Compliance measures how much a unit of volume changes in response to a change in pressure. In this case, the pressure would be the air pressure within the lungs and chest wall. The lungs are compliant because they are elastic. This means that when the air pressure inside them increases, they expand, and when the air pressure decreases, they contract. The chest wall is not as compliant as the lungs because it is not as elastic as the lungs.
The lungs and chest wall compliance are essential because they determine how much work is required to breathe. If the lungs are more compliant, they need less work to expand and less energy to pump air in and out. If the chest wall is less compliant, it requires more work to expand and more energy to pump air in and out.