A prime number is a natural number greater than 1 that can be divided evenly only by 1 and itself. Thus the first few prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
You can see a longer list of prime numbers if you like, or you can play with the Sieve of Eratosthenes, which is an ancient method for identifying prime numbers. The link points to a page with an interactive applet. Using the applet requires that you have a Java compatible browser.
You can use another interactive applet to explore the Prime Number Theorem, the distribution of prime twins, and the Goldbach conjecture.
The reason why 1 is said not to be a prime number is merely convenience. For example, if 1 was prime then the prime factorization of 6 would not be unique since 2 times 3 = 1 times 2 times 3. A number that can be written as a product of prime numbers is composite. Thus there are three types of natural numbers: primes, composites, and 1.
A useful book on working with prime numbers computationally is: Hans Riesel, Prime Numbers and Computer Methods for Factorization, Birkauser Verlag, 1985.
A standard textbook of Number Theory, intended for use in a first course in Number Theory, at the upper undergraduate or beginning graduate level is: I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991.
Another interesting text is: Paulo Ribenboim, The Book of Prime Number Records, 2nd ed., Springer Verlag, 1989.
A major application of number theory and prime numbers is in cryptography. An excellent introduction to this area is: Kenneth H. Rosen, Elementary Number Theory and its applications, 3rd ed., Addison Wesley, 1993, ISBN 0-201-57889-1.
Fine print, your comments, more links, Peter Alfeld, PA1UM