What was the DAMA dark matter detector reporting in 2013 and why does no-one else believe them? I understand DAMA is a dark matter detector but I don't understand how it works, and what it's seeing.
They claim to have a seasonal variation in their signal which is supportive of a certain type of dark matter? I don't understand this either.
Finally other detectors aren't seeing this, and say it's an error in the DAMA experiment. What steps have DAMA taken to find and fix this and has the experiment been checked by others?
 A: Annual modulation
The DAMA detector has observed an annual modulation in the rate of detected events (see early results in Bernabei et al. (2008)). This means that there is a periodic change in this rate, with a period of close to one year. While I'll give a short explanation here, the importance of this is discussed in detail by Freese et al. (2013).
Let's assume that the Milky Way possesses a halo of dark matter particles. These particles should occasionally collide with other particles, including detectors set up to record the energy and rate of these collisions, because of the motion of objects (stars, planets, humans, etc.) through the halo. Earth orbits the Sun, meaning that at different points in the orbit, it is moving in different directions, with different relative speeds. The collision rate depends on the relative speed of Earth with respect to the halo. Therefore, at different points of the year, detectors on Earth should experience different rates of detected events - in a roughly sinusoidal pattern.
DAMA's data fits this annual modulation. A dark matter halo model fits in well with the evidence, while other explanations do not. Furthermore, the nature of the detector supports a variant of the Weakly-Interacting Massive Particle (WIMP) model.
DAMA vs. other detectors
The evidence gathered by DAMA is good - very good. The results have been reported with 9$\sigma$ certainty, which is extremely high. The problem is, other experiments disagree with the findings. Some have found an annual modulation (though with more uncertainty), while others have not. This is the big issue. Science depends on reproducibility of results. So far, nobody has been able to reproduce DAMA's results. DAMA, however, has continued the search, and so far has returned consistent findings.
