Colloquially, peer-to-peer has become a synonym for peer-to-peer file sharing, often with an implied "illegal" at the beginning.
However, technically, the term "peer-to-peer" simply refers to any connection over the Internet that is made directly between two "peers" rather than between a "client" and a "server".
The Internet is designed to such that all the computers connected to it are on equal terms with one another. They are the "peers". However, this principle has been lost among the proliferation of centralised, client-server infrastructure where end-users are relegated to "client" status and "served" by large, centrally owned and operated "servers". Most of the web follows the client-server model, but as you know The Web is only part of The Internet.
Peer-to-peer applications include file-sharing, certain instant messaging, text chat and voice chat programs (including Skype), remote desktop access, the Tor privacy network, decentralised social networks like Diaspora, some commercial video services (it usually says in the small print), and almost anything else you can think of. The possibilities are endless.
I am appalled that your ISP is blocking or charging for peer-to-peer applications. To do so is to destroy a major part of the Internet's functionality and limit its future possibilities. They shouldn't be allowed to call it an Internet connection if they don't allow you full Internet access. You may have heard about the idea that the Internet was designed to withstand a nuclear war, but this is only true when it is decentralised using peer-to-peer systems!
Having said this, I doubt they are blocking ALL peer-to-peer connections, because it would simply be impractical. They are far more likely to be blocking specific application protocols, or even just specific port numbers. If so, they will be able to give you a list of blocked protocols and/or ports, which you can check against the ports/protocols your applications use.