There are two primary types of distortion that occur within a communication filtering system
Amplitude distortion occurs when the amplitude is scaled as it transits the system in question. This is easiest to imagine in a high pass filter causing lower frequency components of a signal to be degraded. In a HPF described by H(f) = jf/(100+jf), frequency components at 500Hz lose about 2% of their amplitude while frequency components at 100Hz lose 29% of their amplitude.
Phase distortion occurs when the phase of a signal is delayed as it transits the system in question. For example, a high pass filter can cause a substantial phase delay for low frequency components of the signal in question. In a HPF described by H(f) = jf/(100+jf), a 27 degree phase delay happens for frequency components at 200Hz while a 45 degree phase delay happens for frequency components at 100Hz.
Distortion can be compensated for using an equalizer. You may be familiar with these from home audio. An equalizer is placed in series with the signal in order to counteract the effects of filtering on the channel such that the channel appears to be free of distortion.
Another point for consideration are harmonics produced by non-linear filtering. A non-linear filter will contain frequency components that were not originally contained within the input. These frequencies will manifest at the harmonic frequencies for the original frequency. These harmonic frequencies can be filtered out; however, some harmonic distortion will still be present due to this signal degradation. The signal harmonic distortion ratio is the ratio of the amplitude of the harmonic frequencies compared to the original fundamental frequency.