Summary

Peer-to-Peer or P2P is a way of sharing files and resources between computers without the need for a server computer. In P2P, every computer on the network is both the client and the server allowing for computers to make requests to other computers to download files and also to accept requests from other computers to send files.
There are two types of peer to peer services, centralized and decentralized. Centralized services use a main server to keep track of where files are located on the network. They direct users to each other through the use of metafiles. Decentralized services use each network node as both a client and server with the use of the provided software to connect to other users..
Centralized P2P
In centralized P2P networks such as Bit Torrent, metafiles are stored on the centralized server. The metafile contains information for the user about what the file is and who has it but nothing from the actual file. The metafile points the user in the right direction to connect to another user to download it. In this set up of P2P the user's computers are still both server and client but only after the user has downloaded a metafile can it connect to another user to begin the download.
Decentralized P2P
In decentralized P2P networks such as Limewire, there is no use for centralized servers and the distribution of metafiles. Instead the user personally searches what files are available from other users using the software provided and can connect to each other by simply clicking download. In this method the users are allowed to connect to each other as both server and client without the necessity of a metafile.

Potential Legal Issues
While not every download or transfer on a P2P network is illegal, they are not all legal either. While many of the files transferred files are completely legal and can be transferred worry free from one user to another, the illegal file sharing is what has grasped the attention of the music industries and media outlets.
With the invention of P2P file sharing also came the ability for it to be abused. Since it makes files easier to share between users than ever before, it also allows for the sharing of products paid for by one user then gotten for free by another. This kind of file sharing is piracy and is illegal. While piracy of files existed before P2P networks, this technology made it easier and faster than ever before to download pirated files.
Many once legal P2P file sharing methods are now illegal such as Limewire and Napster. However, Napster has made a return to the file sharing industry, this time fully legal and now charging for the music it is distributing. Now Napster avoids the chance of being shut down again for illegal P2P file sharing by only providing its users with legal files such as the was iTunes or Amazon does.

Examples
KaZaA
Limewire
Napster
Bit Torrent
E-mule
Tribler
Pando
References to Relevant News Articles
(edited by Joe)
- SSNs on P2P? The Feds found businesses that leaked private information
- FTC Highlights P2P Network Risks
- P2P encryption for mobile is not an technology endorsement, says PCI Council