What exactly is animation canceling?

I wanted to make a thread to discuss what animation canceling is, and why it's important. I believe a thread should be dedicated specifically to the subject because I found myself utterly lost when researching the subject elsewhere. People say it is simply moving to end the last portion of your AA animation. I do this when I play all the time. I move in between auto attacks when my damage connects or when the projectile is airborne, but apparently animation canceling is something more than that. I have looked into orbwalking/stutter stepping, and I think it may be the sameish thing, but when I hear different people describing it, I can't help but think I am missing something huge. I have found myself spending hours researching the subject on multiple different occasions, and I always end up furious that somehow some people understand some hidden exploit magic that gives them an advantage in gameplay. If this is a valid mechanic (not an exploit) then why isn't there a tutorial in game to show you how to animation cancel? I simply don't understand what it is at all, and I've watched hours of video and pages of info explaining it to no avail. I see people attacking and doing combos faster than should be possible normally in videos, and become really disheartened that I can't figure out what they are doing. Some people describe it as the simple act of moving out of an AA to get movement in between AA's (I do this and its simple), other people describe it as this super skill based trick that artificially increases your attack speed and makes your spells cast faster.

What champions aren't viable without abusing this exploit? I know Riven is often cited as an a prime example.

How can someone learn what this is and how to do it?