There are two basic ways of weighing objects. One of them would give the same reading on the Earth or on the Moon, and the other would not. This is because the two different ways measure two different properties of matter. What are they, and which one would give the same answer on the Moon?

I believe that what you are talking about is the difference between mass and weight. Mass is an absolute measurement of how much of something you have. Weight is how much force that amount of mass will generate when placed in an acceleration field. For mass, kilograms is a measurement of mass. A 10 kg object is still 10 kg whether on the surface of the earth, the surface of the moon or in earth orbit. This gets a bit tricky in English units. When someone says that something weighs 60 lbs, they are really talking about pounds force, LBf. There is unit of mass called, surpisingly, pounds mass, LBm. Now to get from LBm to LBf when talking about an object in a gravitational acceleration field, you take the LBm and multiply by the acceleration. For earth that is about 32 ft/sec sec. Then there is a constant to get all the units to work out right. We have defined the value of this constant such that on earth, a LBm will weigh a LBf. If you go to the moon, 60 LBm will weigh 10 LBf. Since you asked about measurement, I would say that in normal circumstance I would measure the weight with a scale and then multiply out the mass. If you were not in a gravitational field (or accelerating field, Einstein showed us that they are equivalent) then there would still be ways to measure mass. For instance, with a piece of string of known legnth and a stopwatch you could tie the string to the object and spin it about you in a circle and measure the force generated. By knowing the length of the string and by measuring the rate at which the object was moving using the stopwatch, you could calculate out the mass of the object.

Or more simply, just accelerate the object to a measured velocity, and then measure the force it took to return it to 0. Since F=ma, you could then calculate the mass. But yes, you have answered the riddle exactly. Nice work.