A wave is a periodic disturbance in an elastic material.  It is a back-and-forth change of some kind that is repeated regularly as time goes on, that spreads out from a source and carries energy with it.

In wave motion, a disturbance propagates through matter, but the matter as a whole doesn’t go anywhere.

When I create a sound wave the air in this room vibrates:  it moves in a periodic fashion, a back-and-forth type of motion.

The air in this room doesn’t move from one location to another, but the disturbance called sound propagates through the air at over 300 m/s.