Here's the set-up:
You have two identical-looking cups, each sitting on an identical pieces of paper.
You apply Newton's first law or motion (an object at rest will stay at rest, an object in motion will stay in motion, unless acted on by an outside force) to remove the pieces of paper, leaving the cups in the same location.
...when you go to do it, one stays nicely in place while the paper is pulled out from under it. The other cup doesn't cooperate so nicely. It slides around with the paper. In fact, it almost seems impossible to remove the paper from under the cup without lifting up the cup.
Why? What happened to Sir Isaac Newton's law?
Inside one of the cups, you've placed a weight (anything massive will do - marbles, pennies, etc.). Objects with more mass have more inertia - they're more resistant to changes in motion. So, the cup with the weight in it has more mass, therefore it has more inertia and is more resistant to moving with the paper (i.e. it has an easier time staying where it is when the paper is removed).
The styrofoam cup with nothing inside of it has very little mass and therefore very little inertia. As a result, it is very difficult to remove the paper from underneath the cup.