I assume you know what a prime number is. There are infinitely many of them!
The following proof is one of the most famous, most often quoted, and most beautiful proofs in all of mathematics. Its origins date back more than 2000 years to Euclid of Alexandria who lived around 300 BC. Euclid's argument was different, but this is the proof that is most commonly given today:
An interesting book on prime numbers is Paulo Ribenboim, The New Book of Prime Number Records, 2nd ed., Springer Verlag, 1996, ISBN 0-387-94457-5. Starting on page 3, it gives several proofs that there are infinitely many primes.
Here are a couple more interesting references on prime numbers:
Fine print, your comments, more links, Peter Alfeld, PA1UM