Question

how is matter measured?

Answer 1

Matter is measured in lots of ways: Temperature, mass, momentum, trajectory, energy, color, amount, whatever.

I am guessing that the question actually is, how is mass measured? Mass is kind of a deep concept, if you think about it long enough. The idea is that stuff accelerates inversely in proportion to the mass under a given force. In old school physics lingo mass was called the inertia constant. If something accelerates less for a given force, it has a higher inertia, or in our words a larger mass. Mathematically represented: F = ma.

Formally this sort of mass is called inertial mass, and technically this is the definition of mass. However on earth we usually measure (measure doesn't mean find out exactly) the gravitational mass. We do this by measuring the force on a scale and dividing it by what we think is the value for gravitational acceleration at the surface of the earth (it's not the same everywhere, but it's close). So there are subtle differences to the word mass, and it really depends on how it is measured. In physics, it is considered a deep and profound subject. Also, Energy=mass*the square of the speed of light.

I bet if you researched Newton's experiments on mass you would get a good idea for the essential question and then know where to look for other mass measurement experiments.