If you connect a computer to another computer, you have a network. If you connect a computer to lots of other computers you also have a network. It is just a lot of computers connected to one another in a way that they can communicate with each other.
The internet is just the biggest network of computers where any computer connected to the internet is part of this global network. It is huge, but it is just a network of computers.
If you are on the internet, your computer is connected to a server (just another computer) which is connected to lots of other servers around the world.