Paxos
Leslie Lamport's consensus algorithm (1989) — lets a group of unreliable nodes agree on a value even with partial network failures.
Two phases: prepare/promise and accept/accepted. A majority (quorum) must respond to both before a value is committed. Multi-Paxos optimises away the prepare phase when the same leader stays stable, which is what makes it usable in practice.
Famously hard to understand — Lamport's own "Paxos Made Simple" paper became an inside joke. Used in Google Chubby and Spanner; most new systems today pick Raft, which is pedagogically simpler but fundamentally equivalent.