Saturation used to be something that happened in the analogue world. Typically, this is when a gain stage is overloaded – the signal level exceeds the available headroom. When this happens, the signal is saturated.
Basically, the sound gets distorted because you turned it up too high.
The result of this is that the parts of the signal that were going to exceed the available headroom are waveshaped. If you were look at the waveform of a saturated signal, you’d see that the loudest parts of the sound have been clamped down – they’re quieter than they should be. This is similar to what a compressor does, except that saturation affects the shape of the waveform itself – not necessarily the perceived volume level (as we hear it). By changing the shape of the wave, the sound changes too.
Saturation reduces the level of transient peaks by distorting them. However, because the transient peaks are very short, the disortion is often not obvious. The excess level in the transient peaks is transformed into upper harmonics. That is, the transients become noisier and dirtier. For some kinds of music, this can be a desirable alternative to reducing gain using a compressor or limiter. The power and impact of the sound is often retained (or even enhanced!), but at the expense of fidelity.
Saturation of steady-state signals is often more noticeable because the audio is constantly being saturated. This usually causes the sound to be brighter, as upper harmonics are being created. Too much saturation will make the audio sound lo-fi or outright distorted. Used subtly though, saturation can make audio sound more exciting, or even aggressive. In a dense mix, individual tracks will sound less distorted than when listening to those tracks on their own (in solo).