The Internet is...
The internet is based on the idea that computers can exchange information in much the same way you talk to people on the telephone. The term refers to the many computers which are connected to each other worldwide, and which are therefore able to communicate.
Computers can exchange information in a variety of formats. Two of the most popular are email (which allows people to send text to each other's computers), and the World Wide Web (or, simply, "the web").
The web refers, actually, to documents (called web pages) which, although they're kept on a given computer, can be viewed by people on other computers. (The computer which has the page on it is called the "server", while the computer that looks at the page is called the "client.")
Web pages, which appear similar to magazine pages, can include text and images, as well as fancier things like sounds and even video. They are written in a language called HTML (Hypertext Markup Language) which describes how the page should look. A "browser" program on the client computer (that's the computer looking at the page) understands HTML and displays the page appropriately. Commonly-used browsers include Netscape Navigator, Microsoft Internet Explorer, Mosaic and Lynx.
One important reason the web has grown so quickly is that it is relatively easy and cheap to publish web pages. Compared, say, to a magazine or television program, a web site (a collection of web pages) can present a lot of information to a very large number of people for very little money. This has made it possible for individuals and small organizations to communicate with a mass audience.
For more info, check out:
Web 101 - Webmonkey's introduction to the web.
Zen and the Art of the Internet - a classic, readable explanation of the net.