Pressure is the first and most important factor that must be overcome for humans to be able to breathe underwater.
Properties of Water:
Water is much more dense than air. This means that water has more mass per unit volume than air and will weigh more per unit volume as well.
Water is uncompressible. This means that water under great pressure (force per area) will have the same volume as water under much less pressure.
What does this all mean? Well, the first property can be felt when you dive to the bottom of a pool. You immediately feel pressure on your ears. This is because all the water above you is pressing down on you and squeezing the air inside your ears into a smaller volume. You do not experience this when going down a short flight of stairs because air is not as dense. The extra air that presses down on you at the bottom of the stairs does not weigh enough for you to feel it.
The second property is important in that it explains why water will have a pressure directly proportional to a specific depth in the water. Since water weighs 64 pounds per cubic foot, the underwater pressure on 1 square foot area will increase by 64 lbs every foot increase in depth. Double your depth, double your pressure, etc.
How all this relates to scuba diving: When you swim down underwater, the high-density water all around you presses in from all sides. This means that is you were to try to breathe through a hose from the surface of the water, your lungs would need to push all that water weight outwards to make room for the air. This is impossible and that is where scuba comes in.
When you breathe from a scuba tank, you breathe air at the same pressure as the surrounding water. This means that, although the water presses in on you, the air you breathe presses back out with the same force. This makes it easy to breathe. Breathing pressurized air under water is the same as you are breathing now. The air around you is pressing in on you from all sides, but it is not hard to breathe because the air you are breathing is pushing out with the same pressure as the air around you.
All images courtesy Corbis Stock Photography. Created in 2003 by David Baxter & Timothy Schmidt.