My wife worked in Clean-up at Disney for years and she once found this old Disney photostat on "Follow Thru" at an animation art dealer.
A couple of definitions: "Clean Up" is the category of artist that draws over the animator's rough drawing and creates the clean drawing that will appear on the screen when the movie is seen in it's final form. This is quite a discipline in itself and it pretty unknown and unappreciated by the general public.
"Follow Through" or "Overlap" is the word for the things in animation that move because a character or object moves but cannot move by themselves. For example the clothes and hair on a character.
This is a photostat from long ago that teaches the basics of follow-through. I scanned it at a high resolution. If this large size is unwieldy so I created a smaller one too (the lower one is the smaller version).
Overlap is one of those things that is a very simple, straightforward concept but can become quite difficult in more complex situations. The best approach (as it is with everything) is to always keep it simple, simple, simple.
There were always some animators who would actually leave it off in their ruffs and let the clean up people add it for them. There were also animators who would leave it off in their initial pass and after the movement, timing and acting were finished they would go back and add it to their drawings. The advantage of this approach is that you're only focusing on the most important things in your first pass and not distracted by unnecessary details. However, I have to say that you could always tell when people animated this way, because each drawing is a design and overlap has to be carefully integrated into every drawing because it's not just a static, mechanical process - just like everything else, overlap has to be used to frame the action and accentuate the acting, it's another tool to convey the feeling and mood of the scene and character - it has to be an organic and essential part of the design of the whole drawing and idea of the scene...if that makes any sense. We all know how great a drawing is when we do it "all in one" and every part relates to everything else. When we do a drawing and then go back later and add a detail to it, the detail usually looks like it was "stuck on" because it wasn't part of the original thought and wasn't considered as part of the original elements and design. So if possible I think it's better to at least have it in mind and rough it in at least minimally in the first pass of the drawings.
However that's probably good advice for some and terrible for others. Keep in mind that I was never a very good or accomplished animator. Maybe some of the very skilled animators out there might give their two cents in the comments so I don't accidentally send newcomers down the wrong path!