What is Dark Matter?

The story of dark matter is best divided into two parts.  First we have the reasons that we know that it exists.  Second is the collection of possible explanations as to what it is.

Why the Universe Needs Dark Matter

We believe that that the Universe is critically balanced between being open and closed.  We derive this fact from the observation of the large scale structure of the Universe.  It requires a certain amount of matter to accomplish this result.  Call it M.
We can estimate the total baryonic matter of the universe by studying Big Bang nucleosynthesis.  This is done by connecting the observed He/H ratio of the Universe today to the amount of baryonic matter present during the early hot phase when most of the helium was produced.  Once the temperature of the Universe dropped below the neutron-proton mass difference, neutrons began decaying into protons.  If the early baryon density was low, then it was hard for a proton to find a neutron with which to make helium before too many of the neutrons decayed away to account for the amount of helium we see today.  So by measuring the He/H ratio today, we can estimate the necessary baryon density shortly after the Big Bang, and, consequently, the total number of baryons today.  It turns out that you need about 0.05 M total baryonic matter to account for the known ratio of light isotopes.  So only 1/20 of the total mass of the Universe is baryonic matter.
Unfortunately, the best estimates of the total mass of everything that we can see with our telescopes is roughly 0.01 M.  Where is the other 99% of the stuff of the Universe?  Dark Matter!
So there are two conclusions.  We only see 0.01 M out of 0.05 M baryonic matter in the Universe.  The rest must be in baryonic dark matter halos surrounding galaxies.  And there must be some non-baryonic dark matter to account for the remaining 95% of the matter required to give Ω, the mass of the Universe, in units of critical mass, equal to unity.
For those who distrust the conventional Big Bang models, and don't want to rely upon fancy cosmology to derive the presence of dark matter, there are other more direct means.  It has been observed in clusters of galaxies that the motion of galaxies within a cluster suggests that they are bound by a total gravitational force due to about 5-10 times as much matter as can be accounted for from luminous matter in said galaxies.  And within an individual galaxy, you can measure the rate of rotation of the stars about the galactic center of rotation.  The resultant "rotation curve" is simply related to the distribution of matter in the galaxy.  The outer stars in galaxies seem to rotate too fast for the amount of matter that we see in the galaxy.  Again, we need about 5 times more matter than we can see via electromagnetic radiation.  These results can be explained by assuming that there is a "dark matter halo" surrounding every galaxy.

What is Dark Matter?

This is the open question.  There are many possibilities, and nobody really knows much about this yet.  Here are a few of the many published suggestions, which are being currently hunted for by experimentalists all over the world.  Remember, you need at least one baryonic candidate and one non-baryonic candidate to make everything work out, so there there may be more than one correct choice among the possibilities given here.

• Normal matter which has so far eluded our gaze, such as:
  • dark galaxies
  • brown dwarfs
  • planetary material (rock, dust, etc.)
• Massive Standard Model neutrinos.  If any of the neutrinos are massive, then this could be the missing mass.  On the other hand, if they are too heavy, as the purported 17 keV neutrino would have been, massive neutrinos create almost as many problems as they solve in this regard.
• Exotica (See the Particle Zoo FAQ entry for some details.)

Massive exotica would provide the missing mass.  For our purposes, these fall into two classes: those which have been proposed for other reasons but happen to solve the dark matter problem, and those which have been proposed specifically to provide the missing dark matter.
Examples of objects in the first class are axions, additional neutrinos, supersymmetric particles, and a host of others.  Their properties are constrained by the theory which predicts them, but by virtue of their mass, they solve the dark matter problem if they exist in the correct abundance.
Particles in the second class are generally classed in loose groups.  Their properties are not specified, but they are merely required to be massive and have other properties such that they would so far have eluded discovery in the many experiments which have looked for new particles.  These include WIMPS (Weakly Interacting Massive Particles), CHAMPS, and a host of others.
References: Dark Matter in the Universe  (Jerusalem Winter School for Theoretical Physics, 1986-7), J.N. Bahcall, T. Piran, & S. Weinberg editors.