most p2p networks are just a bunch of people sitting at their home PC's all running some software program that creates a "network" between them. usually when a user starts this program they will connect to some central server that sends them a list ofa ll the other users logged on.
p2p, or Peer to Peer, is a networking design where each computer holds their own access controls, and users/passwords are managed on a machine by machine basis. Peer to Peer networks are found in many computer environments, such as Windows (Workgroups and Personal File Sharing), Macintosh (Appleshare (although AppleShare can also be a server too)), and Linux ( NFS File Sharing).
The contrast to Peer to Peer would be a Client / Server network (Windows 2K Server, Appleshare / Netatalk Server, and I am not certain what the Linux style is called, and Novell). Username / Passwords are stored on a central server/database, and the clients login to that central administration. This model of design is much much easier for system stability and security.
In reality, p2p networks are fine for simple connections, home use, that sort of thing. Businesses should incorporate formal server environments for system stability, ease of management, and backup simplicity. At my personal home, I run this format, so that all my backups are in one location. Practice what I preach.