What's the application? LIGO stands for Laser Interferometer Graviitational Wave Observatory, because (astro)physicists feel free to drop inconvenient words when making up cute acronyms. This is an experiment to look for disturbances in space-time caused by massive objects, which would manifest as a slight stretching and compression of space itself.
What problem(s) is it the solution to? 1) "Can we directly observe the gravitational waves that are predicted by the equations of General Relativity?" 2) "Can we detect things like colliding black holes, because that would be awesome!"
How does it work? LIGO consists of two detectors, one in eastern Washington state, the other in Louisiana, that are basically the world's largest Michelson interferometers:
Light from a laser is split by a partially reflective surface, and sent along two perpendicular paths. At the end of each path, they hit a mirror that sends them back to the original beamsplitter, where they are recombined and sent to a detector. The amount of light reaching the detector depends on the relative length of the two paths. If the lengths are exactly the same, the light from the two arms will be exactly in phase (peaks lining up with peaks, etc.) and all of the input light will reach the detector; if one path is half a wavelength of light longer than the other (round trip), the waves from the two arms will be out of phase, and no light will reach the detector.
A gravitational wave coming along will cause space to stretch and compress slightly along the direction of the wave, which will cause one of the arms to get a little bit longer, and a little bit shorter. This should show up as a small fluctuation in the intensity of the light reaching the detector, which can be analyzed to give the strength, freuqnecy, and direction of propagation of the wave.
Of course, this is fiendishly difficult to do, as there are lots of other things that might cause changes in the intensity of the light at the detector, from noise in the laser to vibrations of the mirrors caused by things like passing trucks or clumsy post-docs-- that's why they have two widely separated detectors, because random sources like trucks and oafs aren't likely to hit both Washington and Louisiana at the same time. They're also looking for a ridiculously small change in the length-- something around the diameter of a proton-- so the arms are huge, and involve many mirror bounces back and forth along the path. This means that whole Ph.D. theses have been written about ways of minimizing the noise effects of vibrations and the like.
Why are lasers essential? They need to send two beams of light over several kilometers, and then overlap them again. You can do this with non-laser light-- the original Michelson-Morley experiment used a lamp and some filters-- but you really, really don't want to. Believe me. A laser is the best tool for the job if you're setting up a Michelson interferometer on a lab bench, and using anything other than a laser for LIGO is simply inconceivable.
There are also ways to manipulate lasers to use "squeezed light" with a lower intrinsic noise than any classical source, which have been proposed as a component of the next generation of LIGO. They're not using squeezed light yet, but it's something you can really only get with a laser.
Why is it cool? There have not been any confirmed detections of gravitational waves yet, despite the fact that they are predicted by General Relativity. Their presence has been inferred from studies of things like pulsars, but nobody has directly observed the sort of space-time disturbances you expect to see from gravitational waves, and that would be way cool.
The potential sources of gravitational waves are all extremely cool-- things like giant supernovae forming black holes, or colliding neutron stars, and that sort of thing. LIGO is potentially a new window for looking at these kinds of phenomena, which can't be detected other ways. It's even possible that some gravitational wave observatory might detect primordial waves left over from the Big Bang (LIGO won't get there, but other proposed experiments might), which would provide all sorts of information about the shape of the universe, and actually provide data relevant to things like string theory.
Probably the coolest thing I can say about LIGO is that their continuing attempts to reduce the noise fluctuations in their detector have led them to study the cooling of gram-sized objects close to their quantum ground state. This is incredibly cool stuff, and it's a side effect of their real experimental goals.
Why isn't it cool enough? They haven't detected anything yet. It's all noise and no signal so far. Which is more or less what they expect given the current configuration, but it's still way less cool than it would be with a confirmed detection.
Can you tell us the size of the thing, in kilometers, or do we have to google it ?
"3,002 kilometers (1,865 miles)"
As further proof that LIGO is awesome, I submit this music video made by a bunch of undergrads working on LIGO.
The fact sheet I grabbed the picture from (click on the image to get there) gives the arms as 4km in length.
Arthur: 3000 km sounds about right for the separation of the two sites. The detectors themselves are obviously much smaller than that.
I don't know anything about the Louisiana site, but the one in Washington is on the Hanford site. They could build a pair of 4 km tunnels there with absolutely no risk of running into anything. Local lore has it that the Hanford site wound up in federal hands because everybody agreed that even the Indians deserved better land than that.
What's the predicted wavelength for gravitational waves? Somehow gravity related events don't seem to be high speed affairs, and the "antenna" size would need to be huge for something in the 1 Hz range.
The wavelength for gravity waves depends on the source. But the two LIGO observatories are geared towards picking up events such as two neutron stars spiraling in towards each other in their final "death throws." This puts the frequency on the order of 10's of Hz. to the kHz range (around the ranges of hearing)...check out http://www.ligo.org , or http://www.ligo.caltech.edu/ for more information. To detect very low frequency gravity waves there are a couple of other projects...LISA (still a concept - a space based interferometer to achieve the massive distances needed to detect low frequencies), and a binary pulsar-based way of detecting gravity waves.