P2P (Peer-to-peer) is a protocol for 'peer' computers to communicate among themselves to provide solutions to problems that are not solved as effectively or efficiently as client/server technology. A few years ago, the file-sharing service, Napster, gain notoriety for allowing individuals to share music files. The music industry did not like it, and sued Napster out of existence.
Aside from file-sharing there are situations where P2P protocol is the best solution. See http://www.openp2p.com/pub/q/p2p_category for a comprehensive list of links to P2P applications.