What is the mass of our galaxy? Obviously, we cannot put the entire galaxy on scales...
Apparently, one can think of more than one way of measuring it. For example, of all the objects in our solar system, the sun is by far the heaviest. How heavy? The sun is more than 500 times heavier than the combined masses of all the other objects (planets, moons, asteroids...) in the solar system. Thus, the sun alone accounts for more than 99.8% of the total mass in our solar system. The sun is also the only object in the solar system that emits light; all other objects, such as the earth and the moon only reflect it.
Now, astronomers have found a way to estimate the mass of stars like our own sun, based on the light they emit (this is known as the "Hertzsprung-Russell diagram". Of course, there are some uncertainties, but they are not huge. Using this, one can simply "count the light" of all the stars in our galaxy (or any other galaxy), and deduce the galaxy's mass.
However, there is a second way. Any mass, as we know, applies gravitational force on all other objects. This forces cause the other objects - stars in our case, or entire galaxies, to move. The speed of the motion is related to the force that act on the objects, which is proportional to the mass. Interestingly enough, it is not very difficult to measure the velocities of stars and galaxies. This is done using an effect known as "Doppler effect" (after Christian Doppler that discovered it in 1842), according to which waves emitted by a moving object change their frequency (think of the sound from an incoming train).
Naturally, astronomers assumed that the two independent methods of measurements would lead (roughly) similar results. However, it turned out that the results are not similar at all. Measurements using the second method (based on gravitational attraction), revealed that the gravitational force applied by galaxies require approximately 5 times (or 500%) more mass than the amount of mass inferred by "counting the light" method. This was first noted by the astronomer Fritz Zwicky, in 1933.
Thus, it seems that for every kilogram of "normal" mass in the universe, there are about 5 kilograms of "dark" mass, which for this reason is known as "dark matter". This mass does not emit or absorb any light, hence we cannot see it. We can thus conclude that it cannot be made of "normal" matter. However, we can feel its gravitational force, which is inferred on very large (galactic) scales, but not on smaller scales like that of our solar system or here on earth.
Zwicky, in fact, measured the velocities of clusters of galaxies. Since his time, there are several independent pieces of evidence that lead to the same conclusion. Some notable proofs include the measurements of galaxy rotation curves, done by Vera Rubin in the late 1960's. Other evidence come from gravitational lensing - a general relativistic effect, according to which the path of light rays are being "bent" when travelling close to a strong gravitational field. Using this "bending", one can infer the mass that creates the field.
A very famous example is the "Bullet" cluster (pictured above): this is a system composed of two clusters of galaxies, which happen to collide. In the collision, the dark matter was not affected, but proceeded straight away; this can be inferred by gravitational lensing. However, the "regular" matter, such as dust, was slowed down due to, e.g., electromagnetic forces, and was heated. Thus, we can see the regular matter, which is hot, as it emits radiation in the X-ray band. By comparing the gravitational lensing to the X-rays, we can tell that for this cluster, the location of the dark matter is different than that of the normal matter, providing further evidence that these are two separated entities.
Different evidence for the existence of dark matter is found when looking at fluctuations of the cosmic microwave background radiation. The fluctuations "imprint" some of the initial conditions, and from their location and strength it is possible to infer the amount of dark matter present.
While there are several ideas of what dark matter could be, as of today, there is still no compelling evidence for what it really is, and is one of the great mysteries of modern physics.