One of the biggest scientific mysteries of our time is why the galaxy doesn’t simply fall apart. Yeah, that’s right: we know that planets and indeed stars are kept in their orbits by gravity and not some kind of magical force. The problem is that, once you’ve added together the mass of the stars, interstellar gas, dust and all the objects we know about…there just isn’t enough gravity to go around. Not by a long shot: we should actually be seeing about 85% more stuff.
Sneaky Little Particles, Hiding all Over the Place
This is pretty much a certainty: between Newton and Einstein, we have a fair idea of how gravity works. Measuring the mass of a planet or asteroid, which determines how much gravitation it emits, is not very difficult. So, why doesn’t this number match the actual amount of gravity?
The most popular hypothesis among astronomers is that the “stuff” is in fact there, we just can’t observe it directly, hence the name dark matter. Even the name of this website is a kind of inside joke: there really is no such thing as a dark matter telescope.
No one has ever found so much as a cupful of the stuff. We don’t have a clue what it consists of, though some kind of quark soup seems like a strong possibility. Since we don’t know, we might as well call all the matter we can’t observe through radio waves, visible light or other kinds of radiation “dark”.
Detecting Dark Matter
Just about all we understand about dark matter is that it has mass. A little bit of detective work therefore allows astronomers to see its effects. In particular, scientists had suspected for decades, and finally proved it in 1979, that light passing near a massive object is deflected somewhat like the lens of a refractor telescope does:

(A true geek will tell you that the light is still traveling in a straight line; it’s the structure of space itself that gets bent. The more you know…) This means that tiny distortions in the way we see distant astronomical objects give us clues about how much dark matter may lie along the path traveled by its light.
Additional observations tally with these gravitational lensing experiments. Dark matter is one of the easiest ways to explain the shape and size of far-off galaxies: if there weren’t some extra substance of unknown composition in there, they would look very differently, rotating much more slowly or being less crowded. To put this into perspective, our sun orbits the center of the milky way at about 500,000 m.p.h, yet still takes 230 million years to go around once.
Perhaps the most visually interesting evidence for the existence of dark matter comes from observing collisions between and within galaxies. Galaxies are, of course, almost completely made up of empty space, but when stars are forced into relatively close proximity, they do have an effect on one another.

As it turns out, dark matter basically ignores the ordinary kind except when it comes to gravity. Intriguingly, however, it seems that it does somehow interact with itself, occasionally making large clumps of invisible material form and separate from what we can see. These can be detected, including by the methods mentioned above.
A Good Concept, But Far from Certain
While dark matter seems plausible enough for a ton of money and effort to be spent on looking for it, the hypothesis (which is what you call a scientific explanation that agrees with the available evidence, but hasn’t been adequately tested through experiment) is not universally accepted. Some observations can be interpreted in favor of its being true, others seem to argue against this.
What we can conclude is the following: either our theories of astrophysics aren’t 100% complete yet, or something really interesting exists out there. Whatever science fiction has told you about dark matter, the truth is probably stranger. If so, there’s a whole new field of knowledge out there waiting to be discovered.
Related Information:
Deep Space Telescope
A little more about the instruments used to search out cosmological phenomena.
MOND
One of the theories that tries to explain gravitational anomalies without exotic forms of matter.