The Fabfilter article is pretty good... but just in case you want more:
Mid-Side processing is used all over the place in audio.
A front facing cardiod can be paired with a side-facing bi-directional microphone to record both the mid and side signals. This provides almost infinite control over the stereo width during mixing.
Electronically realized mid and side signals are used for FM stereo broadcasting, and were once used to encode stereo material for storage on broadcast cartridges (which were notorious for their terrible stereo performance - the tape would slide up and down and make a real mess!)
And the much respected Fairchild 670 provided a way to create Mid and Side signals for processing to make it easier to cut a record... meaning quite literally cutting the vinyl.
Let's do a little bit of math (just a little!)
The Mid signal, in all of these cases, consists of the mono equivalent, what you'd hear if you put up only one microphone. It can be created, after the fact, but adding the left and right channels (and you will see it referred to as L+R)
The Side signal is created by subtracting the Right from the Left. This is also what you get from a side-facing bi-directional microphone.
So why is it useful?
Mid is monophonic - and compatible with mono radios, which is why it is the primary signal broadcast by FM stations.
If you add Mid and Side you can, in effect, decode the Left channel:
(L+R) + (L-R) = 2L - since the right channels will cancel out
If you subtract Side from Mid you can decode the Right channel:
(L+R) - (L-R) = (L+R) + (-L+R) = 2R - since the left channels now cancel out
That's how stereo FM receivers work. It also turned out that even with the tape slip-sliding around if you encoded the stereo into Mid and Side you could recover a pretty darned good facimile of the original stereo from a cart machine.
None of which is all that valuable to us today.
But what is valuable is we can now work on four different channels of a stereo signal! We have Left and Right, of course, but we also can have Mid, which is the mono equivalent, and Side, which is the ambiance, if you will. (yeah, not a great description, but my brain is off for the weekend.)
So I can apply equalization or compression or delay or whatever to the MID channel and only affect those things that appear in both the left and right channels, the "center" channel if you will. That's pretty powerful! And I can do the same to the side channel, but that's not always as useful<G>. All the while I can also still process or effect the left and right channels.
Clear as mud??
PS- Carver and Polk Audio (among others) used to use this matrix idea to create a "bigger than stereo" effect. It sounds cool, for a little bit anyway.