The fundamental concept in peer-to-peer architecture is that the components of the software exist in multiple computers (nodes) and are interconnected among themselves. Unlike client-server architecture, in this type of structure, a node could take the role of either and contribute to the network as a part of it.
Peer-to-peer is used when a system demands a decentralised structure. Distributed computing and file-sharing is a common use case where it is applied.
For example, as seen in the crash course on cryptocurrency, blockchain uses the P2P network. Another good example is the popular file-distribution protocol BitTorrent that leverages P2P architecture effectively.