A cosmologist's 'dark matter' (non-baryonic) is different than an astrophysicist's 'dark matter' (baryonic). To an astrophysicist, the term 'dark matter' has historically meant matter that is not lit up. It is not reflecting ar emitting light. Also it is not blocking light from some other source. There is nothing exotic or strange about it. It is just in the dark and so it cannot be seen.
There were many observations of matter within the milky way, and within other large spiral galaxies that showed the velocity and orbits of matter were not explained by the mass that could be seen. We only saw mass in the visible light for a long time. The matter had to be emitting light, reflecting light, or blocking another source of light for us to see it in telescopes.
It was simply assumed that Einstein's theories of gravity were still correct and there just had to be more matter than we were seeing. It wasn't seen becuase it was dark, hence the name 'dark matter'. Nothing wierd or strange, just stuff we didn't see.
As time went on our observations expanded into more regions of the electromagnetic spectrum. We saw that there, indeed, was a great deal more matter emitting in the infra-red, radio, x-ray, and gamma ray spectrums. This has added greatly to the amount of matter that is known. There is much less missing mass on the intragalactic scale than there once was because we see more of it.
However, it is not enough. Here is a really good explanation.
And there is a new problem. We are now mapping the interaction of galaxies, and of huge groups of galaxies. And there does not seem to be enough matter in sight to fully account for there movements. Enter the cosmologists.
The first 'exotic' form of 'dark matter' was probably the neutrino. While once considered a very exotic beast, it is now considered rather mundane (at least the three known flavors are considered mundane). The neutrino is an almost massless particle that is electrically neutral and has such a small cross section that it hardly ever interacts with other matter. Neutrinos have mass, so they do feel the effects of gravity and due the the equal and opposite reaction thing, they contribute to the gravity that we, our sun, and all the starts in the galaxy feel. While a single neutrino is almost non-existent, the huge numbers of neutrinos within the boundaries of the galaxy actually do add up to an appreciable mass.
Now cosmologists are suggesting even more exoctic unknown particles, like WIMPS, to explain the missing mass. Some people feel that we should be examining new theories of gravity. Maybe on a very large scale gravity behaves differently. We do know that our theories of gravity are not complete. We do not have a good field theory of gravity that works with quantum mechanics. Continued experimentation involving things like the Higg's Boson will help to confirm some of these leading edge theories, and get rid of others. By determining the mass and energy of the particles that communicate the 'mass' field we will be putting constraints from the real physical universe around these theories.
The cosmology stuff is the wierd exotic 'dark matter' that inspires wierd science fiction ideas, but it will probably be needed to explain all of the missing mass. When some of these, currently, exotic particles are observed measured and fit in an overarching theory, they will seem much more ordinary, as the three known neutrinos are today.