Gravitational waves were first predicted in 1916 by Einstein’s general theory of relativity; today we are trying to directly observe them. A gravitational wave is a tiny oscillation in the fabric of space-time that travels at the speed of light; all other findings from general relativity predict its existence. Many objects will create minuscule gravitational waves, and even the largest objects create ones we just barely hope to see (such as binary stars and black holes). From the LIGO wikipedia page “gravitational waves that originate tens of millions of light years from Earth are expected to distort the 4 kilometer mirror spacing by about 10−18 m, less than one-thousandth the charge diameter of a proton.”
What would we gain from this? Astronomers believe that gravitational waves could eventually become another mode of imaging by which to analyze the universe, like gamma ray, x-ray, and infrared imaging.
The LIGO (Laser interferometer gravitational-wave observatory) ran from 2002 to 2010; it was unsuccessful in its hunt for gravitational waves. It is being recalibrated to restart in 2014. The two observatories in Louisiana and Richland, Washington record the same events and compare the time at which they arrive. Below is a schematic of this set-up. LISA, the laser interferometer space array, has been discussed for years as an orbiting detector with greater length scales (and therefore greater accuracy) than LIGO; a proof-of-concept is due for launch in 2014.
If you want to learn more, Einstein Online, which is run by the Max Planck Institute, is a great resource (the Max Planck Institute is involved in great cutting edge research, perhaps comparable to NASA). The above link is for info on gravitational waves, but there is also great info on other concepts related to relativity if you are interested.