Look what a quick Google search turned up: http://www.isoc.org/internet/history/
Basic summary: the American Research Projects Agency (ARPA) funded a company called Bolt Beranek Newman (now BBN Technologies) to link four universities together to form the ARPANET. This grew for some time and other similar networks were created, including NSFNET and JANET (two academic networks in the US and UK respectively). Cross-communication between networks became possible, and the combined "international network" became known as the Internet. Tim Berners Lee invented the World Wide Web (i.e. HTML and HTTP) as a means of sharing physics research in the early 1990s, and this triggered much more interest in the network.
Another note: the original purpose of the ARPAnet was to share computer time at universities. It was not military, though that came later with MILnet.