I am struggling to understand the difference in audio mixing between stereo separation, delay, and mid/side EQ when it comes to widening a sound. Can someone explain these in laymen terms, specifically stereo separation?
WE humans have two ears, and are accustomed from very early in life into hearing really small details that one ear hears different from the other. Stereo separation is used in editing with different objectives, being mostly creating a sorrounding feeling to what is being listened. There are different techniques, the first one being one sound going through the left speaker and not the right speaker, or difference in the volume level between these two. But there are some other techniques as well that you can look into, like retarding a sound several miliseconds and giving the brain the feeling that the sound comes (spatially speaking) from one side or the other -of course, the side that is reproduced early is the source of the sound where it should be coming. This is using very little time separation (between 0.1 miliseconds (specially in headphones) to around 20 or even more miliseconds). So the delay is closely related, but it is usually used more time separation, therefore creating an even more "wider" sound. Sometimes this is overused sort to speak because in reality you only get some time separation, but you also may take into account the reflections that the walls, ceiling, or floor of the enviorment is directioning the original sound to your ears a bit later than the first wave of sound you heard directly from the source.
All these times separation have different ranges to create different effects. The smallest separation are usually creating a left to right sensation, while repeating sounds with a time separation of more than 40 miliseconds generate a reflection of the enviroment feeling.
Mid/side eq refers to adjusting only the sounds that are being reproduced from both speakers, or only one of the speakers (may be the right or left, but only one of them) So when you select mid, you are only affecting sounds that come from both speakers, and when you select side, you are only affecting sounds which are only being reproduced from one of the speakers.
I also recommend you to using headphones and listening to binaural sounds and looking into that a bit, which takes the surrounding feelings to the limit, making incredible inmersive sounds, very natural to hear. Here is a fantastic example:
Also here's a link with something I myself made with a binaural plugin: Again, headphones are NEEDED:
I think the other answers deal pretty well with the topic of stereo, but if I am reading your question correctly, they miss the mark a little: "stereo separation in the context of mixing" is simply taking the recording(s) of the different instruments and panning them a little left and a little right in the stereo field (as described int he other answers) so that you wind up "carving out" a spatial location for that instrument. This gives the instrument a little clarity over having all the frequencies fighting for the middle.
Another technique that is used to help with clarity is to use EQ to "duck" or lower the other instruments' audio in the main frequency range of the current instrument so that each instrument has more ownership of that particular frequency (i.e. bass owns the low end, the guitar owns a chunk of the mid-range etc.) this lessens the overlaps in frequency range between instruments and provides for a little more clarity.
Obviously, both of these are mostly useful if you have isolated recordings of each instrument.