Many very large airplanes have specific information of their empty weight. For instance the weight of an empty, regular 747-400 is 393,263 lb (178,756 kg) according to Wikipedia. The Concorde is 78,700 kg.
How is the weight of these airplanes measured so precisely (note that in the 747-400 example is precise to the unit's place!). Is the sum of the weights of all the parts used? Also, is Zero Fuel Weight (ZFW) the same as empty weight?
The empty weight of airplanes can only be given to single-pound precision for one specific airplane. Minute differences in metal sheet thickness, equipment and paint will make sure that there is a difference of many pounds between airplanes of the same type and furnishing. Add to this the difference in seating, galleys and instrumentation between airplanes of the same type which are operated by different airlines, and a precise number as in your question becomes almost meaningless. A better way would be to give a range in which most airplanes will fall, or to round the number like the Concorde weight of your question.
Weighing is quite straightforward: All tires of the landing gear are placed on scales and the weights are summed up. If the attitude of the airplane does not change, the weight on the tires can even be measured consecutively, so only one scale is needed. Small floatplanes can be suspended from a crane with a scale in the line holding the aircraft.
In Ernst Heinkel's biography he remembers an aircraft deal with the Soviet Union in the late 1920s. He had to deliver five floatplanes of a specified weight, and the Russian approval inspectors insisted on all planes having the exact same weight. Of course the planes were not identically heavy, but his advantage was that the scale in his factory was indicating the weight by printing it on cards. So the evening before the inspectors came, he let one aircraft be weighed five times, put the cards back into the machine and had the printing mechanism disabled. The next day he closed the business successfully with his very happy Russian guests.
Addressing the ZFW question, zero fuel weight is not the same as empty weight. Empty weight is the weight of the airplane without any people, bags, cargo or fuel on board. Zero fuel weight is the weight of the airplane plus people, bags and cargo, but does not include fuel. The ZFW is important because there is usually a limitation of maximum ZFW. For example, once you hit max ZFW you can no longer add people or cargo to the airplane, even if you would stay under max gross weight including fuel. The only weight you can add to the airplane above the max ZFW is fuel.
You get yourself some load cells, put them under the gear axles / lift points etc. and jack it off the ground. The attached computer will tell you the total mass, center of gravity etc.
Several companies produce aircraft weighing scales.
Be aware that if a small change is made to an aircraft, like adding or removing equipment, the airplane is typically not reweighed but it's empty weight is recalculated by adding or subtracting the weight change. Also, the empty weight c.g. is recalculated by adding or subtracting the moment change.
As I remember, there is also an requirement that aircraft be reweighed every so many years. I want to say every 3 years, but that may be wrong.
Also, the empty weight does not include pilots and cabin crew. The term BOW (Basic Operating Weight) is often used that does include standard weights for pilots and cabin crew.
Air carriers, particularly freight operators, may employ more than one BOW to allow them to easily switch between configurations. For example, freighters sometimes carry fly-away-kits (FAK) of several hundred pounds. The kits are strapped down, but can be quickly removed if need be. The operator might choose to have two BOWs, one including the FAK and one without it.
There's also the matter of using "fleet weights" rather than individual airplane weights, or at least there used to be.
