Excuse the crass headline, but William Shatner posed the question to us a couple days ago. To him, and possibly to you, it's the biggest mystery in the universe. So, to appease his notorious temper, here's a quick-and-dirty answer.
Dark matter is usually explained either by what it is or what it does. Both are pretty interesting. What it is is matter that doesn't feel the same forces that normal matter does. Specifically, that means electromagnetic forces. So, light, electricity, magnetism, and most of the stuff that makes normal matter "normal" and within the normal world of experience. Thus, it's matter that can sail through just about anything and not interact with it.
This is why we look for dark matter deep underground with detectors placed in old mines and the like. Dark matter will cruise through the Earth's crust unimpeded, while everything that might interfere with the search gets filtered out by the ground. One place we look for dark matter is in Antarctica, at a project called IceCube. IceCube is a kilometer-square block of stunningly pure ice that acts as a natural observatory for dark matter (or, rather its probable by-products; we still can't actually see it).
If dark matter doesn't actually interact with anything, what's the big deal? Well, it still has mass, and acts gravitationally. It therefore has huge effects at a larger scale within the universe. Dark matter is actually about 83 percent of matter in the universe, and acts as a force that holds galaxies and galaxy clusters together. Without dark matter, we wouldn't be here.
In 1933, Fritz Zwicky postulated its existence to explain the movements of galaxies. They were behaving as if they had missing mass; some source of gravity from something unseen was helping them out. We've since confirmed it again and again by looking at the rotational speeds of galaxies and using gravitational lensing, observing how gravity "distorts" light on its way to Earth. As of last month, we even have a pretty good map of dark matter in the universe (within about 6 billion light years of Earth, at least).
Now, if you think dark matter is weird, it's still got little on dark energy. Dark energy accounts of 73 percent of everything in the universe, including energy and the energy "within" matter (the energy potential of matter, since they're the same thing). It's even weirder. Dark energy is everywhere in extremely tiny amounts. A cubic meter of space has about the energy of one hydrogen atom. That might not seem like much, but at the galactic or universe scale, it adds up to a lot. Dark energy isn't particles or stuff; you can think of it as the energy of empty space.
Vitally, it has negative gravity, and thus pushes the universe outward. And as the universe gets pushed outward and gets bigger, there's more empty space, so, more dark energy. That means that the dark energy-fueled universe expansion will just keep accelerating, which pretty much means that eventually the universe will be stretched so thin that it will be almost completely dark energy in a field of infinite plainness. As you might expect, many scientists agree that dark energy is the most exciting concept being explored in physics right now.
One important thing to note: everything that is actually "light" in the universe, like us and all of the stuff we can see out there only makes up about five-percent of everything. We're in the vast minority here, which is pretty absurd when you think about it. Imagine if you left your home city just to find everyone else on the planet not only didn't speak your language, but acted like you didn't exist. You would be a ghost. Maybe we already are all ghosts. How's that for a mystery, Shatner?