In light of the recent experimental results at LIGO, I thought a few words on gravitational radiation are in order.
Gravitational waves are a phenomenon which occurs both in linearised gravity, as well as a solution to the full Einstein equations. They are periodic disturbances in the gravitational field, a time-varying gravitational influence; in the language of General Relativity, they are wave-like perturbations of the geometry of space-time itself, and as such do not require any medium of propagation. For the purpose of this short introduction, we shall limit ourselves to the linearised case; the basic idea here is to describe the metric of space-time as the flat Minkowski metric plus a small perturbation :
We can then introduce a rank-2 tensor field ( “gravitational field” ) , which is related to the perturbation via
There is a freedom here to impose a gauge condition; a suitable choice, such as the Lorentz gauge , reduces the Einstein vacuum equations into the particularly simple form
The reader will immediately recognise this as a linear, relativistic wave equation; physically, this means that disturbances in the gravitational field propagate as a wave. The simplest solution to the above wave equation is a monochromatic plane wave of the form
In order for this to be a valid solution, two boundary conditions must hold – firstly, the wave vector must satisfy
which means that it must be a null vector, and hence that the wave must propagate at the speed of light. Secondly, the amplitudes must satisfy
which physically means that the wave propagates in a direction orthogonal to the disturbance itself. The easiest way to visualise what such a wave might “look like”, is to consider its effect when it passes through a suitable ensemble of test particles. Suppose we have a ring of stationary, non-interacting test masses, oriented such that the plane spanned by the ring is perpendicular to the propagation direction of the gravitational wave; the effect of the passing wave will then be a relative acceleration of these test particles, like so  :
The motion here is greatly exaggerated to demonstrate the principle, in reality the acceleration induced will be tiny, even for very strong sources – which is of course why it took so long to directly detect such waves in the laboratory. Once can also plot the above in three dimensions, to show the propagation of the wave itself :
Gravitational waves can be polarised linearly or circularly; in each case, there are two possible polarisation modes. The above two animations depict a linear “plus polarisation”; the second mode, called “cross polarisation”, would be offset by 45 degrees, like so  :
The effects of a circular polarisation are best shown in three dimensions :
The two possible polarisation modes in the circular case are called “left-handed” and “right-handed” – for reasons which should be quite obvious from the above animation.
The source of gravitational radiation is a time-varying quadrupole ( or higher multipole ) moment; this is different from electromagnetic waves, the source of which are dipole moments. Examples of systems with a non-vanishing higher moments would be binary systems, or rapidly rotating bodies which are irregular in shape. In the case of the first LIGO detection, the source was the merger of two black holes with several dozen solar masses each. The signal lasted roughly 0.2 seconds in the detector, and included the initial merger as well as the subsequent ring-down phase. This might be visualised along these lines :
We have up to now only considered linear and purely plane waves, for academic purposes; of course, real-world gravitational waves will be neither linear, nor will they ( in general ) be plane. When we go away from the linear approximation and consider the full non-linear Einstein equations, things become very complicated very quickly. For starters, a more general gravitational wave will not be plane, i.e. it will have both a transverse as well as a longitudinal part. Such a disturbance might look like this :
Furthermore, due to the non-linearity of the Einstein equations, gravitational waves will self-interact; that means they will interact with themselves, with other waves, and also with background curvature. As a result, you can get effects such as refraction and backscattering, even in regions which are otherwise empty vacuum. This, again, is very unlike the electromagnetic case, which exhibits no such self-interactions ( Maxwell’s equations are linear ). Another subtle and important difference between these two types of radiation is that the energy-momentum “carried” by a gravitational wave is not localisable – it does not make mathematical or physical sense to speak of the “energy in a single wave front” . All one can do is write down the effective energy-momentum averaged over several wave lengths.
So why is all this so important ? Well, for one thing, being able to directly examine gravitational waves will give us a new tool with which we can observe the universe around us, and which provide us with new insights not obtainable from the EM spectrum. More importantly though, the wave that was detected had precisely the form and characteristics we expected it to have – this is a strong indicator that our model of classical gravity ( General Relativity ) is indeed a good and valid description within its domain of applicability. The gravitational wave manifested precisely as predicted, and the dynamics of the source ( binary black holes ) likewise is precisely as predicted, meaning it matched exactly the template obtained from numerical calculations of such scenarios, based on General Relativity.
 Misner/Thorne/Wheeler, Gravitation, §35.7, page 955 onwards