In the early days of computers, people used them to compute. Decades later it made sense to use them to communicate and store words and pictures.
By the time computers became cheap enough that it was not shameful to use them as typewriters, most of the computers in the world were connected to a few other nearby computers to form small local networks. These were typically in universities. No university wanted to give up the independence of its computing setup just to connect to other universities, so 'the internet' was created as a set of agreements between university people about how they would connect their computers together.
'I will use the highway to come and visit you'
There could well have been two or three or five internets, but there weren't. There was just one, and once it really got rolling (1978 or thereabouts) everybody who understood it wanted to connect to it. Commercial connections to the internet have been permitted since about 1992, and the internet has been dominated by nonacademic interests ever since.
These days 'the internet' means much more than just the connections between computers. It also means the connections between people, and the access that people have to information. But the internet is new enough that people still talk about it as mechanism. You would find it odd to hear someone say 'I will use the highway system to come visit you in my motorcar.' In California they would say 'I'll come over'; use of the car would be assumed. There is no point in mentioning the use of the highway system for driving: nothing else makes sense. You don't hear 'I'll use the telephone system to talk to you'; rather, you hear 'I'll ring you'.
The mechanism, once it works well enough, becomes invisible, and we talk only of the actions that the mechanism enables. Everyone knows what 'I'll ring you' means. In a generation, phrases like 'electronic mail' or 'web site' will sound as quaint as 'horseless carriage' or 'subscriber trunk dialling'.
Is it a cyber-something?
Fundamentally the internet is the thing that you get when you connect a large number of people together so that they can exchange information with one another, store it, and look at stored information. It isn't really 'cyberspace' or cyber-anything. It's relationships between people, and relationships between people and libraries of data.
The word 'cyberspace' and other science-fiction terms are misleading, because they focus on the mechanism, the wires, the computers, the cyber stuff, and not on what the mechanism can do for you. If I send a note to my mother, you may be certain that I am thinking about her and about what I want to say to her, and not about the wires and machines that will carry my note to her home.
Is it anything new?
The value of the internet is that it enables communication between people who otherwise might ignore each other, and that it lets people get to libraries of stored information that they otherwise might not be able to see.
In the small, this ability has existed for a long time. Small local networks of computers have enabled groups to communicate internally for nearly an entire generation. What is new is the global reach, and global compatibility, of the internet.
The internet is so named because it is an interconnection of networks. It connects small groups to form a single large group. Conceptually there are two parts to internet communication: the mechanism and the audience. The mechanism is more properly called 'online communication' or 'online messaging'. We will call it 'online communication', and by that term we mean sending and receiving electronic mail and making and looking at web pages.
Online communication is enticingly similar to things that have existed for a century. It's a bit like a telephone call, a bit like a letter, a bit like a magazine or newspaper. The truth is that it's not very much like any of them. Online communication is not like any medium that has ever existed before, and trying to fully understand it by analogy will not work. But it is instructive to look at specific ways that online communication differs from earlier media.