Introduction
P2P is a way of filesharing where peers connect directly to eachother with the use of specialised software in order to transmit files. This stands in contrast to online hosters for example where files need to be on a server which offers it for download.
Since there is a good overview of p2p networks on wikipedia, I won't try to give one here.
The main factors that may help you choose a network over another are usually popularity and security. If a network is very popular, you get more chance to find rare files, and will usually get faster downloads. Examples of popular networks are Bittorrent, Kademlia and eDonkey. These networks also provide protocol obfuscation to prevent ISP's from blocking the networks. On the level of security, if you are concerned about prosecution because you are in a country that is repressive towards filesharing, you might want to try networks that provide encryption and anonymity such as Freenet and GnuNet. Both are young however and the software for them is not very userfriendly nor popular yet.
If you are completely new to p2p, bitTorrent is probably the most established and userfriendly system to get the hang. The emule family provides a powerful network with access to a lot of rare files, but usually the clients have a lot of settings to get right to get it working.