Mid/side processing is a different way of processing two audio channels. Most processors modify a stereo sound by applying the same modification to the right and left channel simultaneously. Some processors can have different settings for the right and left channels. Mid/side processors, however, work on the ‘mid’ and ‘side’ channel instead of left and right.
Two-channel stereo (left/right) audio can be transformed into two channel mid/side (and back to stereo) without damaging the audio. It’s a completely transparent (and reversible) process.
The mid channel contains all the audio that is common between the left and right channels. This includes mono sounds that are panned centre and the ‘central’ sound in stereo sounds. The side channel contains all the audio that is different between the left and right channels. This generally consists mainly of ambience (either natural room sound or artificial reverb) and any sounds that are hard-panned.
By adjusting the levels of the mid or side channels independently, the stereo width of the audio can be modified in a clean and natural way.
Interesting things happen when you start applying EQ adjustments to the mid and side channels independently. This allows the stereo width to be widened or narrowed (or even completely collapsed to mono) in different parts of the frequency spectrum. This is particularly useful for complex stereo audio, such as groups, the mix bus or mastering.
For some practical tips, see this post.