With space and time not as rigid background structures, but as dynamical objects (changing as the world changes in and around them), general relativity predicts fundamentally new phenomena. One of the most fascinating is the existence of gravitational waves: small distortions of space-time geometry which propagate through space as waves!
Most readers will have encountered several wave phenomena in everyday life. Sound waves, for instance: a small region of air is compressed, and the fact that its inner pressure is a bit higher than that of neighbouring regions leads to its expansion. This expansion leads to compression nearby, and in this way, the slight surplus in pressure propagates further and further. Such pressure waves are produced when we talk: our vocal cords compress the air around them, sound travels as waves, and these waves are absorbed by our ears when we hear them. In Einstein's case, the situation is somewhat different, but the basic principle is the same: a slight distortion in one region of space distorts nearby regions, and in the end, there is a moving distortion which speeds along at the highest possible speed (the speed of light). Such travelling distortions of space geometry are called gravitational waves.