Free vibrations are when a physical system undergoes and oscillating motion between different states, but is subject to no time-varying outside forces.
Many simple theoretical models are capable of free vibration modes, such as the simple harmonic oscillator ("mass on a spring"). They generally have a natural frequency at which they oscillate, and if they are perturbed from that natural frequency they will move back toward it over time.
In the real world, however, free vibrations are exceedingly rare. Most real-world oscillations are subject to some kind of time-varying outside forces, from subtle (gravitational pull of the Moon) to dramatic (earthquakes). Even far into deep space you couldn't really escape all time-varying outside forces--you'll still get hit by the occasional gravitational wave.
Still, many real-world systems exhibit approximate free vibrations, and thus will exhibit a natural frequency of vibration. If you pluck a violin string it is not quite a free vibration, but as long as you don't jerk the violin around it will be approximately one. And far into deep space you are removed enough from most external forces that for all practical intents and purposes vibrations will be free.
It is also important to understand that damping doesn't automatically exclude a vibration from being free--there are such things as damped free vibrations. The key is that the damping must be velocity-dependent, sometimes called "viscous" damping because it is the same as that caused by vibrating in a viscous fluid such as water. The natural frequency of a damped vibration is less than the natural frequency of the undamped system, by an amount dependent on the magnitude of the damping.