Saturation is a popular way of enhancing sounds and getting them to sound more "real", "lifelike" or "3D". That's one of the reasons tape machines are still held in such high regard. Saturation makes even live instruments sound more exciting.
These days we have access to many plugins that offer various flavors of saturation. One of my favorites is UAD's plugin emulation of the Ampex ATR-102. I also like their UA-610 Tube Preamp plugins.
I try to capture as much color/texture when recording to the DAW but as a mix engineer I'm usually not involved in my client's recording process so I use various methods to get the sound that I feel is needed for the mix. One method is to run individual sounds or instrument sections out to some of your favorite gear and re-record them onto new tracks. I sometimes send sounds out to the live room PA or to guitar amps and then record them (aka re-amping). If you use this method use recall sheets or your smartphone or camera to notate the settings just in case you need to recall the settings to reprint a track. These days due to time and budget constraints I'll often use plugins to achieve the sound I'm aiming for. These methods work great with virtual instruments that sound a little "sterile".
Here's a quick video by UA showing how to use the 610 Preamp plugin with their Apollo Twin Interface while recording acoustic guitar:
and here's another UA video of a recording session done with the actual console the 610 plugins are based on:
There are many different plugins ( too many to list) that offer saturation but as with hardware it comes down to personal preference.
Here's a short tutorial from Matthew Weiss on The Pro Audio Files showing how saturation can be used to enhance sounds in your mix. Use your monitors or a good pair of headphones when watching the video so you can properly hear the audio.