Peer-to-peer (P2P) describes a network where computers connect directly to each other as equals, instead of all talking to a central server. The Bitcoin white paper's title says it right away: "Bitcoin: A Peer-to-Peer Electronic Cash System".
Client-server vs. P2P
When you use online banking, your phone talks to the bank's servers. The bank sees everything, controls everything, and can shut you out. In a P2P network there is no such centre. Every participant, called a peer, can send and receive information directly with others.
How Bitcoin's P2P network works
Each Bitcoin node connects to a handful of other nodes, by default Bitcoin Core makes eight full-relay outgoing connections plus two block-relay-only ones, and accepts many incoming ones. When a node hears about a new transaction or block, it checks it and passes it on to its peers. Within seconds, information spreads across the whole world, like gossip. New nodes find their first peers through DNS seeds and hard-coded addresses, then learn about more from the peers they meet.
Why it matters
Censorship resistance: there is no single server to shut down. Block one node and the information takes another path.
Resilience: the network keeps running even if large parts go offline or entire countries ban it.
Permissionless: anyone can join without asking. Just run the software.
Peer-to-peer payments
"P2P" also describes how payments feel. You can send bitcoin directly to anyone in the world, without a bank or payment company approving it. The blockchain and consensus rules replace the trusted middleman.
Familiar examples
BitTorrent, used for sharing large files, is another well-known P2P network. Bitcoin applied the same architecture to money, and added proof of work so strangers can agree on a single shared ledger.