I think you are describing one of the characteristics of a wave, which is a vibration. A vibration is one complete back-and-forth movement of a particle motion. A good example here would be a clock that has a pendulum that swings back and forth. One back-and-forth motion would represent one vibration of that wave. It would also represent one cycle of the wave.
There are two types of waves, mechanical and electromagnetic. The mechanical waves need a medium through which to work. A medium is some sort of substance that allows the wave to transmit it's energy. Sound is an excellent example of a mechanical wave. If there were no air, we could all scream our heads off, and no one would be able to hear us!
The second type of wave, electromagnetic, has no need for a medium. As a matter of fact, electromagnetic waves work better in the absence of a medium. Radiant, visible light energy from the sun is a super example of an electromagnetic wave. It only takes 8.3 minutes for the radiation from the sun to get to Earth, despite the tremendous distance it has to go to get here.