The term Darknet has been widely used ever since the 1970's when it was used to describe a small network of computers that operated outside the ARPANET network. Since it is not connected to the network it would be "dark" to those on it. However, in some cases darknets were connected to ARPANET but were technically invisible as the addresses did not appear in the network list and the machines would not respond to certain requests, such as PING.
These days, the term Darknet tends to make any informed user think of Peer-to-Peer file sharing networks. Broadly, a darknet can be any collection of friends who have their own network setup, whether it be for filesharing or communications or anything, as long as privacy from outsiders is required. This is why Darknets exist in P2P these days, either setup by friends who have known each other online and trust each other, or often on a College campus to avoid being targeted by copyright infringement lawsuits or simply to avoid having activites tracked.
In P2P, a Darknet is a private file sharing network, so it technically does not apply to networks like FastTrack or Overnet networks, as they are built for public sharing and any user can view shared items and download files from any other (however, some clients do allow you to pick and choose who can share with you).
These days, Darknets have become controversial topics and the Internet is loaded with copyright infringing material that is traded and consumed on a daily basis. To tackle this problem, trade groups such as the Recording Industry Association of America (RIAA) have hired third parties to monitor the activity of file sharers, resulting often in lawsuits (which are usually settled for a few thousand dolalrs). This led to more interest in Darknets, where the RIAA could not monitor activity and in most settings, would not even know that the network exists.
Projects like the Freenet project constitute a Darknet by nature as they are made to ensure private exchange of information. Other such anonymous systems, often made solely for file sharing, are also in development.
Related glossary to this item include..
Freenet's anonymous P2P by years end? - (August 4th, 2005)
An article showing some details on Freenet before the project had any releases whatsoever.