The most popular theory of its nature is the so-called WIMP hypothesis, for Weakly Interacting Massive Particle. Since nothing is known about WIMPs, this doesn't pin them down much, but presumably that arose when the universe was hot enough in approximately equal numbers of particles and antiparticles. After the temperature cooled enough, they went their separate ways, influenced only by gravity.
They cluster under gravitation, and according to most models, ought to produce a spike of concentration at the centers of galaxies and galaxy cluster. Most evidence, though, indicated that that spike is somehow rounded off.
Back to Lumo's articles, first a popular account in Quanta Magazine.
Not long after the Fermi Gamma-ray Space Telescope took to the sky in 2008, astrophysicists noticed that it was picking up a steady rain of gamma rays pouring outward from the center of the Milky Way galaxy. This high-energy radiation was consistent with the detritus of annihilating dark matter, the unidentified particles that constitute 84 percent of the matter in the universe and that fizzle upon contact with each other, spewing other particles as they go. If the gamma rays did in fact come from dark matter, they would reveal its identity, resolving one of the biggest mysteries in physics. But some argued that the gamma rays could have originated from another source.And from the ArXiv:
Past studies have identified a spatially extended excess of ~1-3 GeV gamma rays from the region surrounding the Galactic Center, consistent with the emission expected from annihilating dark matter. We revisit and scrutinize this signal with the intention of further constraining its characteristics and origin. By applying cuts to the Fermi event parameter CTBCORE, we suppress the tails of the point spread function and generate high resolution gamma-ray maps, enabling us to more easily separate the various gamma-ray components. Within these maps, we find the GeV excess to be robust and highly statistically significant, with a spectrum, angular distribution, and overall normalization that is in good agreement with that predicted by simple annihilating dark matter models. For example, the signal is very well fit by a 31-40 GeV dark matter particle annihilating to b quarks with an annihilation cross section of sigma v = (1.4-2.0) x 10^-26 cm^3/s (normalized to a local dark matter density of 0.3 GeV/cm^3). Furthermore, we confirm that the angular distribution of the excess is approximately spherically symmetric and centered around the dynamical center of the Milky Way (within ~0.05 degrees of Sgr A*), showing no sign of elongation along or perpendicular to the Galactic Plane. The signal is observed to extend to at least 10 degrees from the Galactic Center, disfavoring the possibility that this emission originates from millisecond pulsars.It could be big - see Lubos and other links therein. Better test are coming, both in new instruments and by looking at dwarf galaxies, which are rich in dark matter but relatively poor in the sorts of astrophysical competitors that can put out gamma rays.