Let's begin with the observatories. There is one observatory in Louisiana, and two in Washington state. The gravitational wave detector consists of two lasers which go in perpendicular directions. Each laser is 4 km (2.5 miles) long, and encased in a vacuum pipe. Once the lasers have bounced back and forth in their tubes many times, they recombine and interfere with each other. By looking at the interference pattern of the lasers, we can determine the difference in length of the two laser paths. And by that, I mean we can measure the difference very sensitively, down to 10^-18 meters. This is about a thousand times smaller than an a proton.
One of the Washington detectors. Credit: NASA
Why do we want to measure so sensitively the length of a laser path? It all goes back to Einstein.
Albert Einstein is most famous for his theories of Special Relativity and General Relativity. Special Relativity describes how physics behaves when things move near the speed of light. General Relativity is the theory which incorporates both Special Relativity and gravity. In fact, General Relativity is the theory which replaces the classical theory of gravity. The classical laws are very accurate under most conditions, but are decidedly incorrect nearby very massive objects and when things are moving near the speed of light.
In a way, it's rather surprising that General Relativity and classical gravity could possibly be describing the same thing. Classical gravity describes everything in terms of forces. General Relativity describes gravity as a distortion of the space-time topology. In other words, gravity influences the distances and time-intervals between different events. These small distortions cause a straight line through time appear to be curved, as if it were acted upon by some force.
One of the predictions of General Relativity is the existence of gravitational waves. Gravitational waves are analogous to electromagnetic waves (aka light). Electromagnetic waves are fluctuations in the electric and magnetic fields. Gravitational waves are fluctuations in the space-time topology. Electromagnetic waves are created whenever an electrically charged object accelerates. Gravitational waves are created whenever a massive object accelerates. Both kinds of waves are characterized by a frequency, which tells you how quickly the waves fluctuate. If a gravitational wave passes through the LIGO detector, it will cause the two laser arms to fluctuate in length. If the gravitational wave has a frequency of 40 Hz, then the lengths will fluctuate 40 times per second.
LIGO is only sensitive enough to detect gravitational waves with frequency 40 Hz or higher. At lower frequencies, it becomes too difficult to distinguish between gravitational waves and regular old earthquake activity.
What could possibly cause a gravitational wave of more than 40 Hz? Gravitational waves are caused by accelerating massive objects. For example, the earth is constantly accelerating towards the sun because it is in a circular orbit. But this should only cause gravitational waves with frequencies of about 1 per year. However, we might be able to detect orbiting objects if they are orbiting much faster than the earth. One of the objects we are interested in is the binary black hole* system. Black holes are very massive objects, and also very small. So two black holes could be orbiting very quickly and closely to each other. If a pair of black holes is what it takes, then let's look for black holes!
*It could also be any other type of massive astrophysical compact halo object (MACHO), like a neutron star.
One other thing about gravitational waves, is that they carry energy, just like light does. As two black holes orbit each other, they emit energy in the form of gravitational waves. This causes the black holes to slowly lose energy, falling slowly towards each other. Because they're closer together, the "force" of gravity is stronger, and they orbit faster and faster. The picture we have here is of two black holes, spiraling around each other, getting closer together and moving faster. Eventually, they collide, coalescing into a single black hole. When there is only one black hole left, it no longer emits gravitational waves, and its signal disappears.
This could really use some animation. So I found some animations on the net from the Numerical Relativity Group.
The detection of gravitational waves is not only a way to test Einstein's theory of General Relativity under new conditions, it is also a new way to do astronomy. It's much like how we build telescopes to detect electromagnetic waves from far away sources. We can use gravitational waves to detect objects like binary black holes, as well as exploding stars, and a certain kind of pulsar. Scientists are also trying to detect something analogous to the cosmic microwave background radiation, only it would be cosmic gravitational background radiation. It would be very difficult to detect, but it comes from a very early point in the universe's history, far earlier than even the microwave background radiation.