Fuel flow can be measures as either a mass flow or a volume flow. Mass flow is preferred, as the calorific value of the fuel relates to its mass. Older engines, say as on a Handley Page Victor, used a primitve venturi fuel flow system. (Fuel run through a venturi, with a pressure of the fuel in the venturi sent to a pressure capsule, with pipe pressure in the outer chamber. The difference in pressure is related to the volume of flow.) Accuracy is about 2%. Other older types might use a variable orifice flow indicator.
Slightly more modern aircraft might use a free rotating turbine placed in the fuel flow, with a magnetic insert in the turbine. As the turbine rotates in the flow, the magnet spins about and an induction coil can measure the amount of rotation - however these don't cope well with high flow rates, as the error increases proportionately. They still only measure volume. All of the above can of course be fitted with a calibration device to convert the volume to a mass, if the density is known.
True mass flow indicators, like a Stator Torque Mass Flow Meter, swirl the fuel in the pipe, and then run the swirling fuel into a spring loaded turbine. The more the fuel can deflect the turbine, the more momentum, and thus true mass any given unit of fuel possesses. There are other similar systems, and these are used on modern turbines (as I said above) so that the Fuel Control Unit can demand an accurate mass (and hence acurate amount of energy) for the engine to use.