In simplest terms, the Internet (capitalized, as it is a proper noun) is a specific collection of computers and the phone lines linking them. This network includes personal computers, supercomputers, and everything in between all around the world.
Okay, but what is the Internet really?
The word "Internet" is very often misused by the media, the general public, and even those who are knowledgeable about computers. It is frequently used as a shorthand way to refer to many of the services and resources available because of the existence of this vast computer network. Consider, for example, the following passage from Ludwig Wittgenstein: (citation forthcoming)
Imagine a foreigner whose grasp of the English language is not yet perfect. You tell her that you will spend the day showing her the University. The two of you go to the campus and walk around, and you show her the campus, the students, the libraries, the classrooms, the faculty, and all of the buildings. At the end of the day, she says, "I've seen the campus, the students, the libraries, and lots of other things, but you didn't show me the University!"
'Internet' can be used to refer collectively to the many resources -- such as the World Wide Web, USENET, and gopher -- that use its network, just as 'university' refers collectively to the campus, students, libraries, and so on.
The next page will touch on each of these (and other) facets of the Internet briefly, with links to more specific information about each topic.
You are, in fact, "on" the Internet right now. The computer you are using is hooked up to a network (probably your campus' local network) that is in turn hooked up to a larger network. These smaller networks are just small pieces of the Internet.
The computers connected to the Internet do what any other computers do: store and process information. Because the Internet is a worldwide network, being connected to the Internet allows you to use, store, and process information on computers all around the world.
Okay, but what does that mean in simple terms?
Communication. The Internet facilitates two basic kinds of communication: static and interactive.
Many of the computers on the Internet contain information that does not change, and much of that information is publicly accessible (i.e. can be accessed from other computers on the Internet.)
As an analogy, consider a library. Think of the computers on the Internet as sections of a library, with the individual files on those computers as books in the library. The information available on the Internet is as wide-ranging as the information in a library, and much of it can be accessed from any computer in the network.
But, like a library, there is a lot of information that any one person might find uninteresting or useless. Thankfully, some of the data stored on the Internet acts like the indexes or catalogs in a library.
Since data can travel along the network at the speed of light, the Internet is also used for direct communication between people.
Most commonly, this takes the form of electronic mail. Other common ways of exchanging information this way include USENET, more commonly known as newsgroups (a collection of subject-oriented bulletin board-like forums in which individuals can contribute to ongoing discussions or debates); Internet Relay Chat, true realtime conversations among groups of people (much like a conference call, only with typing); and, still somewhat experimentally, live-time video and audio broadcasts.