What is a gravitational wave?
A gravitational wave* is a concept predicted by Einstein's theory of general relativity. General relativity states that mass distorts both space and time in the same way a heavy bowling ball will distort a trampoline.
When an object accelerates, it creates ripples in space-time, just like a boat causes ripples in a pond (and also similarly an accelerating electrical charge produces an electromagnetic wave). These space-time ripples are gravitational waves. They are extremely weak so are very difficult to detect. Missions like LISA or LIGO hope to spot gravitation waves detecting small changes in the distances between objects at set distances; satellites for LISA and mirrors for LIGO. As the strength of the wave depends on the mass of the object our best hope of detecting gravitational waves comes from detecting two black holes or pulsars collapsing into each other.
Gravitational waves have been inferred from watching two pulsars spinning and noticing they are slowing down, due to losing energy from emitting gravitational waves.
Gravitational waves are important in telling us about the early universe. The cosmic microwave background gives us a snapshot of the universe about 380,000 years after the start of the universe. Looking very closely at the cosmic microwave background there are patterns seen which can are also be measured in the large scale structure of the universe (so galaxies and clusters) today. These patterns in the cosmic microwave background were caused by very tiny random perturbations from the time when the universe expanded rapidly, known as inflation.
Inflation should also generate gravitational waves. These waves affect the polarization (the way the wave oscillates) of the cosmic microwave background. Measuring the strength of the polarization due to gravitational waves gives us a ballpark figure of the amount of energy involved at the time of inflation and helps pin down when inflation occurred.
*Not to be confused with a gravity wave (which is a wave driven by the force of gravity).