It sounds like you've got the mechanical process by which air enters and leaves lungs down (diaphragm, intercostal muscles). During inhalation, your rib muscles and diaphragm contract, and your abdominal muscles relax. Now for the pressure part of it: The air pressure in your lungs decreases as a result of the pleural cavity expanding, causing air to rush in. Exhalation is opposite: your diaphragm relaxes and your rib cage moves down, increasing air pressure in your lungs and forcing air out. Think of exhaling like squeezing an open balloon.
Why it works is due to the partial pressure. In a mixture of gases, like air, partial pressure of any one gaseous component is defined as the pressure exerted by just that one gas in the mixture. At sea level, the partial pressure of oxygen is about 160 mm Hg, and that for carbon dioxide is only about 0.23 mm Hg, which is much lower, clearly! But by the time this air is moisturized in the trachea and bronchioles, the oxygen becomes diluted to a certain extent. Not too much, but a little less than the values in the atmosphere.
The big deal is the actual exchange of the gases oxygen (to fuel cellular respiration in your body's cells) and carbon dioxide (to get rid of the waste from cellular respiration). This exchange doesn't take place between your lungs and the atmosphere; rather, it takes place across each alveolus in your lungs. The alveoli are each surrounded by a network of capillaries.
Blood arriving at the alveolus in the pulmonary capillaries has a lower partial pressure of oxygen, (about 40 mmHg) and a higher partial pressure of carbon dioxide (about 45 mmHg), than the air in alveoli (100 mmHg for oxygen, less than 1 mmHg for carbon dioxide), which came from the atmosphere. Therefore, since gases always diffuse from an area of higher partial pressure to an area of lower partial pressure, oxygen diffuses into the blood from the alveoli, and carbon dioxide diffuses into the alveoli from the blood.