The foundations of the Internet were formed when packet-switching networks came into operation in the 1960s. Transmitted data is broken up into small packets of data, sent to its destination, and reassembled at the other side. This means that a single signal can be routed to multiple users, and an interrupted packet may be re-sent without loss of transmission. Packets can be compressed for speed and encrypted for security.
Computers at the time were massive, primitive structures. The only type of network in operation before was made up of terminals that logged into mainframes. This is similar to the present-day client/server relationship we have with the modern Internet, except the computers are usually comparable in terms of power, and so the Internet is known as a peer-to-peer system.
Here is a documentary "History of the internet", which is an animated documentary explaining the inventions from time-sharing to filesharing, from Arpanet to Internet. The clip was made by Melih Bilgil.