If you are interested in how P2P applications communicate over the internet, it really depends on which P2P application you are talking about. All of the P2P applications I am aware of transmit over TCP/IP considering it is the foundation that the Internet works on top of.
Each application communicates over a network protocol which is usually specifically tailored by whomever developed the application in question. For example, BitTorrent usually transmits over a range of ports (6881-6889) and the functionality of BitTorrent in regard to how it communicates over the internet is best summed up by the following (taken from WikiPedia)
A BitTorrent client is any program that implements the BitTorrent
protocol. Each client is capable of preparing, requesting, and
transmitting any type of computer file over a network, using the
protocol. A peer is any computer running an instance of a client.
The same can be said for any application, be it server-client based or P2P or whatever other type of topology it uses. All communication happens via network protocols that both computers understand.