Intuitively I think it makes sense but can somebody explain very clearly to a non-physics person why it is that when you pick up a cup from a stack of solo cups very quickly, the cup below it seems to be get "stuck" in the enclosing cup; It is strange because the direction of force needed for them to be "stuck" is the opposite of the direction that results from your lifting. I'm guessing it has something to do with air getting into the enclosing cup, but I'd like to know the specifics.
I'm guessing that you're talking about the stacks of disposable plastic cups. If you lift the top cup quickly you'll often find one or two cups below it pull out as well, and if you then hold the top cup still, the cups below will slide off.
As you guessed, it is to do with air getting into the space between the cups.
The diagram above shows two cups stacked together. When you pull the cups apart you increase the volume of the air between the cups (shaded blue), but to do this air has to flow through the thin gap between the walls of the two cups. Because the gap is small, air can only flow in quite slowly. Because the air can only flow in slowly, if you pull the top cup fast you decrease the pressure of the air between the cups, and this decreased pressure lifts up the second cup.
If you hold the top cup air will eventually flow in through the gap between the walls and the pressure between the cups will rise back to normal. Once this happens the lower cup will fall off.