I'm going to start at the very beginning and try to be exhaustive. There may be readers who are totally new to this concept! So let me try to start at square one, just for fun...
One of the first concepts you hear about when you're studying animation is "squash and stretch", which is the idea that living forms (and certain types of inanimate objects) have a certain amount of flexibility and that they change shape as they move around under their own power, react to external forces or change expression.
The bouncing rubber ball is, of course, the first thing most people approach as an animation test to learn about squash and stretch. Without requiring any complicated drawing, it illustrates how a soft mass can stretch out when it's affected by gravity and squash down when it comes into contact with a hard surface.
Then you learn how to apply this to living forms. As people move from place to place and perform actions, their bodies, limbs, even their clothes and hair (and everything else) are affected by the forces exerted by the muscles working against gravity, wind and air resistance, and this is expressed by drawing the forms squashed down or stretched out. Also, bodies can squash or stretch if they're affected by external forces out of their control (like a person tripping and falling face first on the sidewalk...their face might very well squash into the hard pavement).
Then, there's the more subtle affect of squash and stretch to emphasize an attitude....a character sitting down on a stool, dejected, can sound like a straighforward and static pose, but using a little extra "squash" in the drawing can make him feel like gravity is affecting him more than usual and he has an extra heavy weight on his shoulders...whatever he's depressed about, it's almost like it's physically pressing down on him. Conversely, if he has an attitude change and suddenly brightens up, you might use some stretch in your drawing to show him perking up and escaping gravity a bit. He's lighter on his toes than usual because he's suddenly thought of something that lifts his spirits - a way out of the problem that was troubling him (and squashing him down) a moment before.
The third way I think squash and stretch is really important in animation is to show emotions and thought. There wouldn't be any acting in animation at all without squash and stretch. Squash and Stretch is the only way, really, to show a change of attitude on a character's face and body to show that they're thinking and processing ideas and emotions. When you go from a Squash to a Stretch, or vice versa, you see a clear change of attitude that shows a change is happening within the character's mind.
This last one is one that I find people under utilize in storyboarding, to be honest. If you ever find yourself storyboarding a sequence (or animating one, for that matter) and the viewer isn't quite connecting with the character, and isn't totally clear on what's going on inside the character's mind - what they are feeling and thinking - maybe you're not using the shifts in Squash and Stretch properly to clue the viewer in to what emotional changes the character is going through.
It doesn't have to be an extreme change. Even very subtle shifts of a Squash to a Stretch (or vice versa) can say volumes. We all have seen an actor (or animated character) lift an eyebrow, or purse their lips, or just slump their shoulders, and immediately with that subtle change you know exactly what that character is thinking and feeling and it's more powerful than any more extreme change would be.
Anyway, more to come next time.....