An oscillation is a periodic motion. In other words it is a solution of equations of motion that is a periodic function? of time? with a fixed period . One also talks about slight generalizations of this definition, like damped? oscillations and so on.
A wave is a transportation of oscillations in medium, in other words a disturbance of a medium periodic in time (for a fixed point) and propagation in physical space?.
Cf. quasinormal mode.