First postulated in 1916 by Albert Einstein, gravitational waves are disturbances in the fabric of spacetime. They are caused by, and radiate outward from, a quadrupole (mass distribution) with an accelerating moment. An example in nature of such a system, one that might generate detectable gravitational waves, is a pair of neutron stars in decaying orbit around their center of gravity. The orbital decay results from the loss of energy being radiated away as gravitational waves.
The effect of these passing gravitational waves would be to strain (stretch and squeeze) space. The magnitude of this strain would be in the order of 1 part in 10-20. This is roughly the ratio of sub-atomic and interplanetary distances.
Several projects are currently trying to directly detect these waves. They use infrared laser interferometry along one or two multi-km vacuum tubes (made effectively even longer by bouncing the light back and forth several times) to measure local disturbances in space with (hopefully) sufficient accuracy. Low sensitivity runs have been completed (data analysis is ongoing), and high sensitivity plans are underway.
Detection of gravitational waves would open up a new avenue of astronomy. Firstly, such quadrupoles would emit gravitational radiation even though they might not emit electromagnetic radiation (eg some black hole configurations). Therefore some objects that were previously unobservable would now be visible. Secondly, unlike electromagnetic radiation, gravitational radiation is not absorbed or scattered significantly by matter between us and the observed object. Thirdly, the amplitude of such waves diminishes only by the inverse of distance, not by the inverse square of distance as with light intensity. Thus, much more remote views would be possible, perhaps all the way back to the big bang. Such detection would also be a strong positive test of the General Theory of Relativity.
If such detection does not happen, the implications are equally seismic. In a similar fashion to the observations that required the replacement of the Newtonian model, non-detection of gravitational waves would require the replacement of the Einsteinian model.