What causes waves to be in motion? We know that waves didn't stop , they keep on travelling .
Now its known to us that for a thing to move we require force ! 
Now my question is : Is there any force which causes the motion of the waves ?
 A: Waves don't necessarily travel. There are two kinds of waves, standing waves, and travelling waves. Standing waves have fixed points which they jiggle about, whereas the entire wavefront of a travelling wave moves in time while jiggling.
Next, we don't require a force for something to move if its alrady moving (think back to Newton's First Law, an object in motion stays in motion). In the case of waves, this means they don't require a force to keep them propagating, unless there is something slowing them down, or absorbing their energy.
For light waves in a perfect vaccum there's nothing to scatter or absorb them, so they can travel forever with no driving force. On the other hand, for sound waves travelling through air, the waves give up some of their energy to the air molecules, so eventually the sound waves get attenuated. This is why distant noises are less loud.
A: Yes, the medium the wave propagates through behaves like a collection of particles connected by springs. The energy travels through matter by compressing the springs and moving the next piece of material. Waves can actually stop. For example, sound cannot travel through space--there is no matter to propagate the sound wave.
A: to enlarge a bit on the answer by @ryanlance, to launch the wave in motion requires us to put some energy into the system of masses and springs that he describes, in the form of an initial squeeze to one of the springs or an initial velocity to one of the chunks of mass. Since energy is conserved in wave motion, the wave will then tend to travel off on its own- carrying that energy with it as it does- unless other forces act on it as it moves. Those forces can be dissipative, as pointed out by @coffeecrow, in which case the energy in the wave will get rubbed progressively down to nothingness.
