A Venturi Meter is a device that allows flow rates through pipes to be calculated by measuring the difference in pressure created by a contraction in a pipe. When the flow goes through the contraction it must speed up, and so the pressure must drop. By measuring the two pressures, engineers can directly calculate the velocity of the fluid. Knowing the pipe diameter, this velocity can be converted into a flow rate.
Venturi Meters work based on the principles found in Bernoulli's equation. Here, Bernoulli's equation is set so each term is in units of length, allowing an engineer to calculate the "head" on a pipe. Remember that for a pipe without energy losses, H is constant.
If we can measure the elevation of each pipe segment (or if it does not change), and the pressure head using a Piezometer (A small tube with an opening flush with the wall of the pipe), than we have one equation with two unknowns.
If the pipe diameters are known, conservation of mass law, will give us a second equation allowing us to solve for velocities (or flow rates):
I made this little Mathematica demo (You might need a special plug-in) which allows you to play with a pipe to see how this might work. The yellow parts show a fluid element in the pipe. Notice when the pipe contracts, the fluid element must stretch out (accelerate) in order to fit through the pipe. The demo also shows the Energy Grade line (EGL) and the Hydraulic Grade Line (HGL). Think of the EGL being the total energy at a pont in the pipe. If you stuck a pitot tube into the pipe, the water would rise to this line. The HGL is how high the flow would rise in a piezo tube, and considers just the first two terms of Bernoulli's equation. If the Pressure drops too low, Cavitation might occur!
Feel free to play around with the sliders, and have fun.
For more information on how Venturi Meters work, I'd recommend looking at this youtube video:
If you have any specific questions about Venturi Meters, How they work, or Fluid Mechanics in general, please feel free to ask below. I'd be happy to help.