Large, extended symmetrical lumps of matter behave gravitationally as if all mass were concentrated at the center. So for example, the gravitational acceleration at the earth's surface is what you would expect if all the earth's mass were concentrated at its very center. Likewise, a galaxy's gravitational field is exactly the same as if all its matter were concentrated at the core. Consequently the stars and gas clouds revolving about the core would all orbit at speeds determined by the distance from the center--the farther out, the slower they travel, just like the planets in our solar system. But actual observations show that the galaxy's rotation is not Keplerian, in fact, it seems to rotate more like a rigid body, as if most of its matter were concentrated way outside its visible boundaries. This is the so-called "dark matter".

Observations of clusters of galaxies show that they are individually moving so fast that the clusters should have dispersed long ago. There must be additional, invisible mass (dark matter) in the cluster that helps keep it from flying apart.

In both of these cases, we can calculate (using Newton's Laws) how much dark matter must be there in order to explain these effects, which allows us to come up with estimates of what proportion of the matter in the universe is dark.

My own suspicion is that we haven't really gotten to the bottom of this story. There is something else going on we haven't stumbled on yet.