Saturation is a mild form of dynamic and harmonic distortion typically associated with bringing 'warmth' and 'body' to a sound.
The term relates to a process which can occur in transformers when the magnetic flux generated by the input signal fully magnetises the transformer core such that it cannot accurately pass any larger signals, resulting in audio compression. A similar effect can occur with magnetic tape heads and tape itself. These saturation effects can now be emulated electronically and digitally, often with user controls to fine tune the effect characteristics.