The Internet, with a capital "I," refers to the network that began its life as the ARPAnet and continues today as, roughly, the confederation of all TCP/IP networks directly or indirectly connected to commercial U.S. backbones. Seen up close, it's actually quite a few different networks-commercial TCP/IP backbones, corporate and U.S. government TCP/IP networks, and TCP/IP networks in other countries-interconnected by high-speed digital circuits.
A lowercase internet, on the other hand, is simply any network made up of multiple smaller networks using the same internetworking protocols. An internet (little "i") isn't necessarily connected to the Internet (big "I"), nor does it necessarily use TCP/IP as its internetworking protocol. There are isolated corporate internets, for example.
An intranet is really just a TCP/IP-based "little i" internet, used to emphasize the use of technologies developed and introduced on the Internet on a company's internal corporate network.