Now, what happens when the client machine starts having the processing power, memory, storage and connectivity that we used to associate with servers? We can begin to make these relationships more parallel between machines. An early user of this model is Napster, which uses the term community to define the relationship between peers. Computer A has a number of songs, and User A wants Song X, which he finds on Computer B. User B wants Song Y, which she can find on Computer A, Computer C, or any number of other computers. All of the computers are of the same level, serving songs to those who want and downloading songs from those who have. (Upload and download are terms weighted toward ClientServer, so perhaps a better term is crossload. I won't hold my breath for expanded usage.)
In truth, email, with crosstalk between servers providing the required message-passing, was an early (probably the first) PeerToPeer networking protocol, but it came up a long time before the name came. --DaveJacoby
I suspect that in the early days, computers were expensive enough that all networking was PeerToPeer --- having a "client" implies a cheaper computer. You can certainly consider a glass-TTY running on a serial line to be a very lightweight client computer (most of them had microprocessors, after all), but people rarely do. (WimL?)
See also PeerToPeerWiki.