It is said that it took Einstein ten years before he discovered the way the gravitation worked (since he had to find a way that did not transport gravitation faster than the speed of light he began a search for it). When he eventually found the answer, it was that gravitation was like waves moving through the universe. He also proved that the gravitational waves had to have the exact same speed as the speed of light. How did he prove that? How did he even prove that gravitation is waves? Is it like longitudinal waves or transverse waves? And is it a kind of electromagnetic radiation with a certain wavelength (the waves would then have to have to have a momentum pointed in the opposite direction to that of the wave in order preserve the law of momentum)? How do these waves in that case really work on the objects they're affecting?
Let's see what I can do with this.
Originally Posted by TriKri
You can use the equations of GR (General Relativity) to set up the problem of a mass moving (either with or without a constant velocity) in an otherwise uniform space-time and we see that the gravitational effect on another mass travels at the speed of light. (Essentially the key here is that distortions in space-time travel at the speed of light, since the distortions themselves only transmit energy. They don't have any mass-energy of their own.)
As to the wave nature of gravitation, I believe that Einstein had no (firm) reason for saying this. One of his main motivations in developing GR was an analogy with Maxwell's equations for electro-magnetism, which (at the time) was by most considered to be a wave phenomenon. Now, Einstein himself helped contribute to the theory of the particle nature of light. So there still existed the possibility that gravity might take the form of a particle as well. If it did, since the influence of gravity travels at the speed of light, the graviton would be massless. In this case GR predicts that the graviton has a spin of 2, rather than the spin 1 case for the photon.
As to how such waves work, they are almost identical in properties to light waves. They are transverse waves, have a wavelength, etc. The difference is that such waves couple to every kind of particle, whereas light waves only couple to charged particles. (Graviational waves also cause time distortion, which is a whole other barrel of apples and I don't know enough about this to even speak of it, really.)