The Internet (notice the capital 'I') is and inter-network (or internet, notice the lower case 'i') of computer systems. Right now, while you are connected, your computer is part of the Internet.

The Internet began as a network to connect universities and research organizations together. The US Department of Defense wanted a way to help the defense researchers to share information. Other networks like the National Science Foundation's (NSF) network were created. The NSF network connected other educational institutions. These small networks grew and joined with other small networks to become the Internet. Eventually government funding was reduced and the Internet was commercialized. Now companies like MCI, IBM and Sprintlink provide the services that were once funded. This commercialization caused an explosion in the number of computers and people accessing the Internet. This is where you came into the picture.