A study of sinusoidally forced oscillations of a fractional oscillator shows that the system exhibits a rich variety of damping characteristics. While some aspects of the damping mimic the characteristic features of a damped harmonic oscillator, there are others, which do not find any parallel in the damped harmonic oscillator system. It is clearly demonstrated that the ''free'' and ''forced'' oscillations of a fractional oscillator are characterized by different damping parameters. While both depend on the fractional index α, the ''free'' oscillation damping depends on the ''natural frequency'', ω 0 , of the oscillator, the ''forced'' oscillation damping depends in addition, on the ''driving frequency'', ω. Furthermore, there is a different power-law tail associated with each of these cases.