On the face of it, the lungs are just two big airbags - oxygen in and carbon dioxide out, simple. But within the simplicity, there's a wonderful complexity that we will discuss here. 
To begin, let's follow the journey of air moving its way into the lungs. 
In through the nose/mouth - it doesn't matter which - they join up at the back of the throat. Down past the vocal cords into the trachea (the windpipe).

Half of the air goes down the right bronchus, and half down the left. The right lung has three lobes - upper, middle and lower. The left lung has two - upper and lower - as it has to accomodate the heart.

The airways start wide but get narrower and narrower the deeper we go, from bronchi, to bronchioles, and finally we reach the minuscule alveoli. An alveolus is a mini-airbag in the shape of a bunch of grapes.

When air fills it, it expands - making the skin of the grapes even thinner. This helps the oxygen to hop across into the bloodstream, which sits in capillaries just on the far side of the grape-skin. Carbon dioxide makes the opposite jump into the alveoli.
When we exhale, the air makes the reverse journey, aided by the elastic recoil of the lungs. 

So...how do we breathe? This is where the diaphragm comes in. The diaphragm is a dome-shaped wreathe of muscle at the bottom of the lungs. When we inhale, it contracts and flattens out, this encourages the ribcage to widen. The change in pressure causes a vacuum which encourages air to pour in via our nose or mouth. 

To breathe through your nose, you simply raise your tongue to the roof of your mouth, sealing that entry point. Don't believe us? Try breathe through your nose and mouth at the same time...you can't!

To exhale, the diaphragm relaxes, becoming the original dome shape once more, and the ribcage narrows. This, along with the elastic recoil of the airway, expels the air. 

All this air coming in and out is fine and dandy, but it means nothing without the exchange of oxygen and carbon dioxide gas - that is where the blood supply comes in. Let's follow the bloods journey. 

Oxygen-depleted blood has returned to the right atrium from all around the body. It moves into the right ventricle before being squeezed out into the pulmonary trunk. From here, half goes right, and half goes left - into the pulmonary arteries.
The pulmonary artery carries the oxygen-depleted blood into a lung before it splits into smaller and smaller tributaries, much like a river, spreading all over the lung. The tiny lung vessels are called capillaries, and they sit really close to the alveoli. When the alveoli expand with air, their walls become really thin - thin enough for oxygen to jump into the capillaries and for carbon dioxide to jump out. 

Why does one jump in and one out? Ready, science nerds? - 'Diffusion'. Basically, if there's loads of gas molecules in one corner of a room, they'll move to where it's less busy.
So the oxygen-rich air will always jump into the oxygen-deplete blood, and the carbon dioxide-rich blood will always jump into the carbon dioxide-deplete air. 

Once the exchange is made (now, I sound like The Godfather), the now, oxygen-rich blood in the capillaries will flow into bigger and bigger vessels to eventually form the pulmonary vein. 

The pulmonary veins (left and right) return oxygen-rich blood into the left atrium, ready for the left ventricle to shoot it around the body. 

To conclude - the lungs are like mobsters, designed to make an important, life or death trade. The airways and the vessels need to work in tandem. One is useless without the other. They need to bring oxygen into the body, and remove carbon dioxide. 

Airways job: Fill the bronchi, bronchioles and eventually, alveoli with oxygen-laden air. Empty the lungs of carbon dioxide-laden air. 

Vessels job: Transport the oxygen-deplete and carbon dioxide-rich blood to the capillaries. Get close to the alveoli. Make the exchange, and return the oxygen-rich blood to the heart.

The lungs also do a few weird and wonderful things, like producing a substance 'ACE' (Angiotensin Converting Enzyme), which is important for controlling your blood pressure - but who has the time to get into such things!

The last thing worth mentioning is the pleura. The pleura are sacks that contain the lungs, one on the right and one on the left. The pleural space is the gap between the lungs and the chest wall. Sometimes air or fluid can accumulate in this space, compressing the lung and causes shortness of breath or pain.