Definition
Bernoulli's principle is a law of physics: when a flowing substance (a fluid, such as a gas or a liquid) speeds up, the pressure inside that fluid goes down. Worked out by the 18th-century scientist Daniel Bernoulli, it tells us that where air or water moves fast, it pushes on its surroundings less.
Why does pressure drop when wind blows harder?
Hold two sheets of paper side by side and blow hard through the gap between them. Common sense says the wind should push the sheets apart. Surprisingly, they snap together instead. This is often given as an example of Bernoulli's principle. The usual explanation is that in the narrow gap, the air speeds up, and there the air pressure drops below the pressure of the surrounding air. In reality, the stickiness (viscosity) of the air flowing along the paper's surface plays a part too.
In science, anything that flows freely, like air or water, is called a 'fluid.' A fluid always tends to move from high pressure to low pressure. When your breath lowers the pressure in the space between the sheets, the higher-pressure air on the outside pushes the two sheets firmly inward.
Don't flip the order around, though. Speed isn't eating away at pressure. For a fluid to speed up, the pressure pushing from behind has to be greater than the pressure ahead. So if you follow a single stream of fluid, the places where it speeds up always go hand in hand with lower pressure.
The magic of flow, hiding all around you
A perfume bottle or spray bottle works thanks to Bernoulli's principle, too. When you squeeze the handle, air shoots out at high speed across the top of a narrow tube. The air pressure around the tube's opening suddenly drops.
Meanwhile, the surface of the liquid in the bottle is still feeling normal air pressure. So the atmosphere pushes the liquid up the tube, toward the spot where the pressure is lower. When the liquid shoots out the top, the fast stream of air hits it and breaks it into a fine mist of droplets.
The curveballs and sliders a baseball pitcher throws follow the same idea. A fast-spinning ball has one side where the air meets the spin head-on and slows down. On the other side, the spin and the air move in the same direction, so the air flows very fast. The pressure drops on the fast side, and the usual explanation is that the ball curves toward the lower pressure. This is called the Magnus effect. (The pressure difference is the same thing as the spinning ball bending the air behind it to one side, just seen from another angle.)
A bit more precisely
At its deepest root, Bernoulli's principle comes from one of the great rules of physics: conservation of energy. A fluid flowing through a closed pipe carries three kinds of energy: potential energy from its height, kinetic energy from its speed, and pressure energy from the fluid itself. If friction can be ignored, the sum of these three stays constant along a single stream.
If the pipe is level, so potential energy doesn't change, then whatever kinetic energy the fluid gains by speeding up, its pressure energy must drop by the same amount to keep the total the same. In other words, speed and pressure play an energy seesaw: when one goes up, the other goes down.
A common misunderstanding is that Bernoulli's principle alone explains the force that lifts an airplane wing (lift). But why the pressure differs above and below a wing can only be explained by also looking at how the wing bends the airflow downward. Explaining lift by the pressure difference and explaining it by the action and reaction of the wing pushing air down are not two forces added together. They are two views of the same lift.
Common misconceptions
The lift that raises an airplane wing comes only from Bernoulli's principle.
Bernoulli's principle can be used to calculate the pressure difference above and below a wing. But the same lift can also be explained as the reaction (Newton's laws) to the wing bending the airflow downward. The two explanations are not different forces. They are two ways of looking at the same thing.
Where you meet it
In a typhoon or a sudden gust, the pressure above a building's roof drops, and the roof can be torn upward and blown away.
When two ships sail close and side by side, the water flowing between them speeds up and its pressure drops, so the ships tend to be pulled toward each other.
In one sentence
When a fluid speeds up, its pressure drops, and this happens because of the conservation of energy.