In the movie "Crouching Tiger, Hidden Dragon," Michelle Yeoh's character drops a tea cup and Zhang Ziyi's character catches it.

I have seen characters in other martial arts movies catch cups in mid-air.

What is the concept(s) behind catching a falling cup?

YoungIn Park

A demonstration of perceptual speed and matching reflexes I would think.

Especially useful in a nation without paper towels.

Drop your bodyweight completely (by pulling your feet up simultaneously) so that you move at speed of gravity -- same speed as the cup (so the relative speed between you & the cup are the same)-- then grab the cup (if it's in reach, of course). I've done a lot of this sort of thing in practice & otherwise (esp. since my left arm/hand was injured & I became a bit more clumsy) -- the hardest part for me is the pain in my knee(s) when, mission complete, I stop my bodyweight from continuing the fall (old & new knee injuries love to say "Hello!" during such maneuvers).
Here's the practice (did I post this before?):
(1) Have a partner who's facing you hold a stick (or whatever you're comfortable catching or allowing to hit the deck) -- held at about your belly button/hip level, with one of your hands raised to about solar-plexus/chest level.
(2) Partner releases stick allowing it to drop at speed of gravity -- without warning.
(3) You fetch stick using method described above.

I guess fetching a stick is about as stupid as walking around in a circle...

can't teach an old dog new tricks,
YoungIn Park

Hey YoungIn Park,
If letting ALL falling objects hit the ground is smart, then this fetching exercise would be a dumb waste of time, eh?.

depends on what is in the cup, where it will fall to, and how motivated you are to catch it.

i.e. a cup of 100 dollar bills, hanging over an endless abyss, and how bad you want that x box 360.(or ps3, whichever you prefer)