At right are the results of a numerical simulation of a black hole/black hole collision and the gravitational radiation which is produced (much as the manner in which water waves are generated by disturbances and in the manner in which EM waves are generated by electrical charges).
To get a feel for some of Einstein's ideas, we must start thinking about the space-time of the Universe. Suppose that I tell you that Astronomy 122 meets in 100 Willamette Hall. Is this enough to get you to class? Well, no, because I didn't tell you when the class meets, namely, 18:00-19:50 on MTR. In order for you to show up for class, you must know not only where it meets but also when it meets. This is true for all events in the Universe; you must know not only where the event occurs (its spatial position) but also when the event occurs (its temporal position). The space and time positions are equally important and we should think about events in the Universe in terms of their space-time positions.
This is analogous to what mass does to the structure of space-time. It causes a depression to form so that if an object rolls toward it, it falls into the pit and is captured. (This, by the way, is how Einstein envisioned how gravity works. Mass distorts the space-time causing particles to roll toward the mass. Note that the objects follow the shape of the space-time and in this sense are following an unforced motion! That is, there is no gravitational force, objects are simply following their natural motions.)
Return to the rubber sheet analogy. If I drop a bb on the sheet and it bounces, ripples in the sheet are produced which propagate away from the disturbance. These ripples in the space-time are referred to as gravitational waves.
Graivtitational waves almost certainly exist, but there have been no direct measurements of their existence (there is only indirect evidence that they exist).
Possible sources for gravity waves are supernovae, merging neutron stars or black holes, unstable rotating neutron stars, ... .