In astrophysics and cosmology, dark matter is a postulated form of matter that acts like normal matter when it comes to gravity, but does not interact with normal matter in other ways (like collisions). Apparently, the amount of visible stuff in galaxies (stars, gas, black holes) isn't nearly enough to account for the ways galaxies behave gravatationally, so researchers basically made up this new sort of matter, that no one had ever seen, in an effort to make the numbers work.
And it looks like they were right. One of NASA's orbiting telescopes, the Chandra X-ray telescope, was used to measure the concentration of hot gas in a pair of nearby galaxies, while another optical technique was used to estimate the density of stars. The two galaxies had collided and passed through each other relatively recently. The really cool thing is that dense things, like stars, act like dark matter and don't lose velocity when galaxies collide, but non-dense things, like gas, do lose velocity in the collision. That means that if you can measure where this dense stuff is, and also where the non-dense stuff is, and they don't line up, and additionally if the dense stuff isn't nearly dense enough to account for visible gravitational effects, then you've basically proved that dark matter exists.
The blog Cosmic Variance has a really nice detailed explanation of this research, complete with pictures and a link to this video that shows a simulation of what the researchers think happened. Go read it. It's really easy to understand, and it's one of the more significant astrophysics results in recent years.