Pi is the ratio of a circle's circumference to its diameter. Huh?!? When clothed in mathematical mumbo-jumbo, the concept can sound confusing. John Reid's animated definition of Pi (above) says exactly the same thing, but the animation makes it so much clearer.
The graphic itself was the January 20, 2007 Wikipedia featured animation of the day, but is usually found on the main Wikipedia article on Pi.
The animation also demonstrates Pi's unintuitive nature. The distance across the wheel is relatively easy to approximate at a glance, but it still seems incredible that the distance around it can be over 3 times the diameter! If you want a better idea of how deceptive this can be to people, you can even use this concept to win bar bets! Yes, Pi is actually useful in the real world.
The real world is one thing, but it has also been the subject of much play. For example, you can search for your birthday in Pi. Back in 1995, Mike Keith rewrote Edgar Allen Poe's classic poem The Raven as a Pi mnemonic (Note: My original link to the poem is no longer available)! He called it Poe, E.: Near A Raven. He later incorporated into a second, longer work which he titled Cadaeic Cadenza. I highly recommend taking the time to read the latter in full, even if you have to break it up into several sessions. His handling of the Feynman point is particularly ingenious.
In short, take Pi seriously when needed, have fun with it otherwise.