Waves are oscillations that carry energy. Waves can move through matter or vacuums, depending on what kind of wave they are. Waves that move through matter are called mechanical waves, and waves that move through vacuums AND matter are called electromagnetic waves.
Oscillation: An oscillation is an up-and-down or back-and-forth motion.
Vacuum: A vacuum is space that has no matter in it. NOT like a vacuum that is used to clean rugs.