Air Pressure: Plungers!

On a large scale:

Get your hands on two plungers - new and clean would be preferable (I've actually seen small ones, perfect for this demonstration, at a dollar store)! 

Match up the edges and push the plungers together.

Now try to get them apart.  Not an easy task! 

You pushed all the air out of the middle, so there's nothing pushing them apart.  Only lots of air on the outside pushing them together.

In the end, if you have a good seal, you'll need to force some air in to break the seal, but splitting the plungers apart.

On a smaller scale:
You can have your students perform this activity themselves using suction cups (the kind you sometimes use to hang things in your windows).  The physics is exactly the same, just smaller!