- Gravitational waves are often incorrectly called "Gravity waves". See that page for a discussion of this point.
Gravitational waves are distortions in space that travel at the speed of light away from a mass that accelerates. Predicted by Albert Einstein's Theory of Relativity in 1916. Gravitational waves were first observed and their existence confirmed in early 2016.
Gravitational waves are essentially "ripples", similar to the ripples made in a pond when a pebble is tossed in, but on a cosmological scale. The mass in question are binary systems - pulsars, neutron stars, or black holes - which orbit around each other. According to Einstein's General Theory of Relativity, both objects emit gravitational waves while losing energy as they gradually approach, eventually resulting in a collision at one-half the speed of light, forming a single object as well as emitting a final, very strong burst of gravitational waves
As part of his relativity theory, Einstein came up with his "quadrupole formula", which describes the rate of wave emissions from a system of astronomical masses based on the change of the mass itself, what he called the "quadrupole moment". His formula as originally postulated was
- is the gravitational wave
- is the mass quadrupole moment.
The results of his findings were published in 1916 as Näherungsweise Integration der Feldgleichungen der Gravitation ("Approximate Integration of the Field Equations of Gravitation"), but serious errors led him to a revision, published in 1918 as Über Gravitationswellen ("About Gravitational Waves").
The LIGO observations of 2015
Because gravitational waves from distant (billions of light years away) phenomena are incredibly faint, detecting them has been a daunting task. Decades of work have gone into the development of sufficiently sensitive detectors. In 2015, the LIGO detectors (one in Hanford, Washington and one in Livingston Louisiana) made the first detection of the waves. 
The event of September 14, 2015
This event was observed shortly after turning on the enhanced version of the LIGO detection system, and reported in January 2016. It was the merger of two black holes, each about 30 times the mass of the Sun, about 1.3 billion light years away and hence 1.3 billion years ago. Making sure that it wasn't a detection of random noise required comparing the waveforms with the predicted waveforms from theoretical calculations of the final approach and "ringdown" of a merger of two black holes. The signal closely matched theoretical predictions. The match showed a confidence of better than "five sigma", that is, a probability of less than one in a million that this was a coincidence. The same event was detected at both observatories, a small fraction of a second apart, as the wave passed through the Earth.
Unlike most astronomical wave phenomena, the gravitational waves from black hole mergers are approximately in the audible range. With only a small amount of signal processing, the signal was made audible; it's sort of a "chirp". For a while this "chirp" became something of an internet sensation.
The event of December 26, 2015
Here is a comparison of the "chirps" from the two events. The December event was significantly less powerful that the September one
The LISA project
An even more ambitious and sensitive detector, previously called LISA and now called eLISA (Evolved Laser Interferometer Space Antenna), is being designed by an international consortium of agencies. This one involves satellites in heliocentric orbit, using laser interferometers to measure changes in the distance between "test masses" inside the satellites.
A test of the underlying test mass technology, called the LISA Pathfinder test, succeeded on June 6, 2016.
- Theory of relativity
- Counterexamples to Relativity
- Essay:Rebuttal to Counterexamples to Relativity
- Einstein, Albert. "Näherungsweise Integration der Feldgleichungen der Gravitation"; Proceedings of the Royal Prussian Academy of Sciences (1916), Berlin, Germany.
- Einstein, Albert. "Über Gravitationswellen"; Proceedings of the Royal Prussian Academy of Sciences (1918), Berlin, Germany.