Since it's detected in a negative sense---not visible in gamma rays,
X-rays, ultraviolet, visible light, infrared, millimeter, or radio
regimes, and it doesn't block light either---it's a theoretical happy
hunting ground.  First, let's list some things that can't make the
dark matter.  Most forms of gas are excluded, because atomic hydrogen
would be seen in 21cm radiation, and hot gas would be seen in X-rays
and/or distort the spectrum of the CMB.  Cold molecular gas is a
possibility, but it would tend to collapse into visible stars.
"Snowballs" made of solid hydrogen would evaporate due to the CMB, and
larger snowballs would leave too many craters on the Moon or be seen
as high-speed comets.  "Rocks" are unlikely because there haven't been
enough stars to make the heavy elements.  Faint red stars are excluded
because they're not seen in deep images e.g., the Hubble Deep Field.

This leaves two main classes of dark-matter candidate: large objects
called MACHOs and subatomic particles, some of which are called WIMPs.

MACHOs stands for Massive Compact Halo Objects; examples are
"interstellar Jupiters" or "brown dwarfs," which are lumps of mostly
hydrogen less than 0.08 Solar masses; objects this small don't get hot
enough to fuse hydrogen into helium, and so would be extremely faint
and hard to find.  Other varieties of MACHOs are dead stars, such as
old white dwarfs or neutron stars, and black holes.

The second class is some form of sub-atomic particle; if so, there'd
be millions of these passing through us every second, but they'd
hardly ever interact with normal matter, hence the term "weakly
interacting massive particles" or WIMPs.  Many varieties of these have
been suggested; the only one of these that certainly exists is the
neutrino, but neutrinos may not have any mass.  The number of
neutrinos made in the Big Bang is similar to the number of CMB photons
(few hundred per cm^3), so if they have a small mass (around 30 eV = 6
x 10^-5 electron masses) they could contribute most of the dark
matter. However, computer models indicate that galaxies form much too
late in a neutrino-dominated universe.  Another possibility is the
"axion" which is a hypothetical particle invented to solve a strange
"coincidence" in particle physics (called the strong CP problem).

The most popular WIMP at the moment is the "neutralino" or "lightest
supersymmetric particle"; supersymmetry is a popular way to unify the
strong and electroweak forces (also known as a Grand Unified Theory),
which has some (tentative) experimental support.  Supersymmetry
predicts an unobserved new particle or "superpartner" for every known
particle; the lightest of these should be stable, and lots of them
would be left over from the Big Bang. These probably weigh about
30-500 proton masses.

An important piece of evidence here is "primordial nucleosynthesis,"
which explains the abundances of He-4, Deuterium, He-3 and Li-7
produced a few minutes after the Big Bang; in order to obtain the
observed abundances of these elements, the density of baryons (i.e.,
"ordinary" matter) must be Omega_baryon ~ 0.02--0.1. Since Omega_stars
~ 0.01, there are probably some dark baryons, but if Omega_total = 1
(as inflation predicts) most of the dark matter is probably WIMPs.