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On Earth, we usually determine the mass of an object by placing it on a scale or balance. The weight of an object in Earth's gravitational field lets us determine the mass. But we can't put the Milky Way on a scale. Another difficulty with massing our galaxy is that there are two types of mass. There is the mass of dark matter that makes up most of the Milky Way's mass, and there is all the regular matter like stars, planets, and us, which is known as baryonic matter.
We have several approaches to determining the total galactic mass, which usually involves measuring the speed of things such as stars, globular clusters, or nearby galaxies. Each of these approaches have strengths and weaknesses, though they all give a total value of a trillion solar masses, give or take a few hundred billion.
All of these methods, however, only tell us the total mass. They say nothing about how much of the galaxy is baryonic mass. While baryonic mass is only a fraction of the total, it is what gives us all sorts of cool things like star formation, planetary systems, and digital watches.