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Basically, the Bernoulli principle says that as speed of the fluid increases, pressure decreases. The photo on the right shows this happening. The air in the wide part of the tube has a higher static pressure than the thin part. For a steady flow, the amount of fluid entering the pipe must equal the amount leaving the pipe, so the fluid speed in the thin part must increase.
The full version of Bernoulli's principle includes both the work performed by the pressure and by the changes in potential energy from changes in height. In this form, the principle says the sum of the pressure, kinetic energy, and potential energy is a constant. (Bernoulli does not take into account viscosity or compressibility.)