One may also refer to this effect as soft-clipping. It originates from analog hardware machines and is now commonly emulated in plugins. It is caused by overdriving the input signal, which becomes non-linear to the output. Saturation is a form of distortion that audio engineers use to increase the harmonics and warmth of the track.
The 3 Best Free Saturation Plugins 2021.Top 13 Saturation & Tape Emulation Plugins 2021.