The P2P paradigm has grown beyond its days of file sharing. One of the most successful stories of deployment of P2P paradigm is of Skype.
SETI @ home is also a very interesting project which relies on P2P architecture for sharing computing resources
BitCoin is another one which was already listed.
In the security realm, many of the latest botnets also use P2P communication for 'command & control'. Have a reading about Stuxnet, Conficker (its latest versions), TDL- 4, Storm and Zeus botnets.
basically all applications that may be executed distributed across different hardware resources can benefit from p2p technology. Especially when the input data can be
cut in independent chunks that may distributed (like for example in SETI@home). Depending on the network connectivity it might be beneficial for the performance if the individual components of the application do not need to communicate after they received their chunk of data or their instructions what to do. Commonly these kind of applications is called embarrassingly parallel.
In contrast, real parallel applications, e.g. where the different components of the application are constantly communicating using the Message Passing Interface (MPI), would not be suitable for loosely coupled systems like resources in a p2p network.