How much do we know about dark matter?

We don't know a lot about dark matter, mainly because all of our theories about it have not been supported by observational evidence that can be conclusively identified as dark matter.

The idea of dark matter originated with observations of galactic rotation, and recognizing that the gravitational effects being observed were much greater than the gravity that could be accounted for from normal matter. The "dark" aspect of dark matter is that it doesn't interact with the electromagnetic spectrum in any way; it only interacts with gravity. This might seem like the theory is being made up to fit the facts, but this is the only way to get observable predictions from something you can't see or touch.

The current theory, and the main pursuit of dark matter research, is that dark matter is composed of a particle, or particles, called WIMPs, which stands for weakly interacting massive particles. These particles are hypothetically "relics" from the early universe, basically closer in composition and properties to the "primordial" nature of the universe before it cooled and particles that we're more familiar with began to form. Several observational methods have been proposed, such as looking for interactions between WIMPs and normal matter, generating new WIMPs in a particle accelerator, or looking for signs of WIMPs being annihilated during particle collisions in space. To date, none of the experiments have provided conclusive evidence that WIMPs exist and have properties exactly correlated to this or that - in fact some of them are contradictory, such as the relatively new LUX experiment, which had no results under high-sensitivity observations whereas less-sensitive experiments produced hundreds of results. 

There's at least two other possibilities; the observations and calculations of gravitational effects are in error, or that another property of the universe is responsible for the gravitational difference being observed. Both have been considered, but the prevailing theory is that dark matter or something to that effect is responsible for the observations.