When you plug in an electric guitar into an amplifier, you hear a distinct mode of clipping, which sounds musical and more dynamic than the raw guitar sound. This is known as saturation, but for most guitarists, the lexicon is distortion, or gain. It creates an ordered, uniform clipping that is pleasing to the ears. Saturation is not just for guitars; it is also for creating better mixes.
In a master mix, you would not use a guitar amplifier of course. For many generations, audio engineers have used different equipment to amplify the signal, and these equipment added some desirable artefacts, which brought about many signature sounds for most albums recorded during those dates.
Tube compressors, equalizers and mixers were very common during the early era of music recordings. These tubes added a warm, bright saturation that is not too harsh and its clipping highlighted certain sounds from the recording, or added more desirable qualities to the sound. Think of it as a polaroid camera versus a modern, high-resolution DSLR; it has a distinct quality that is desirable.
Tapes also added a distinct level of warmth and brightness to recordings. Most guitarists prefer tape saturation because it highlights the high-mids, areas where distorted guitars usually penetrate the mix. However, it does this with a distinct, desirable character, rather than just raw distortion.
Saturation could just help you nail the best mix you could do with your current gear. Tube and other saturation plugins, which are free, could be helpful in your audio production.