Gravitational waves are ripples in the fabric of spacetime produced by accelerating massive bodies. Their existence was predicted by Albert Einstein in his general theory of relativity.
In general relativity, gravity manifests itself as massive objects bending the structure of spacetime. In addition, something else happens if the gravitational field varies, for example when two massive objects orbit each other.
The motion of massive bodies through spacetime perturbs its very fabric, imprinting a signal that travels away as a disturbance to the structure of spacetime itself: gravitational waves. The animation visualises the effect of these oscillations, which consist of sequential stretches and compressions of spacetime, rhythmically increasing and reducing the distance between particles as a wave propagates through the surroundings.
It is thought that gravitational waves are abundant across the Universe, typically produced by powerful sources such as supernova explosions and pairs of orbiting black holes. However, despite the attempts of ground-based experiments to detect them directly, gravitational waves so far remain elusive.
(Update 12 February 2016: high-frequency gravitational waves, emitted by a pair of merging black holes, were directly detected for the first time with the Advanced Laser Interferometer Gravitational-Wave Observatory.)