Vibration is similar to oscillation, that is a movement, around an equilibrium point, of an object in a regular manner which exists during a certain amount of time.
The difference is that in oscillation, the energy transforms from kinetic to gravitational potential energy and vice versa, whereas when an object vibrates, the energy exchange is between kinetic and elastic potential energy.
Thus, in solid mechanics a rigid body oscillates (see Newtonian mechanics), while a non-rigid one (elastic) can also vibrate.
