Communicating or interacting with equals.
In networking terms PeerToPeer means that each node can potentially communicate directly with any other node.
The TCP/IP protocols were designed to allow any machine to communicate with any other machine, but these days the increasing use of NAT, dynamic IP address allocation, and draconian traffic filtering is creating a class of "consumers" who can initiate connections to servers but not receive incoming connections. This makes it harder to have peer-to-peer communcations such as voice/video chat without going through a central server, which by definition is not peer-to-peer.
Often used to refer to PeerToPeer file-sharing networks/software.
The problem with pure PeerToPeer Networks is that it can be very hard for newcomers to find peers to communicate with. Generally the solution is to have a few very well known nodes which newcomers can connect to and explore the network from, but this solution immediately privileges the few well-known nodes and they cease to be peers of the other nodes. This can be partly offset by having a very large number of well known nodes, often all the known stable nodes when a release is produced.