The hashgraph consensus mechanism is covered in full under Hashgraph, which explains the data structure, its history and where it is used. This page gives a short definition and explains how the mechanism relates to the network best known for it.
Hashgraph is a method for reaching agreement among computers without arranging transactions into a single chain of blocks. Each node regularly shares, or "gossips", what it knows with randomly chosen peers, and each exchange is recorded as an event that points back to earlier events. The result is a growing directed acyclic graph of who told whom what, and when. Because every node ends up with a copy of this history, it can work out how others would vote on the order of transactions without any votes actually being sent, a technique called virtual voting. The algorithm is designed to be Byzantine fault tolerant, meaning it reaches agreement as long as fewer than one-third of participants are faulty or dishonest.
The algorithm was devised by Leemon Baird and is used by the public network Hedera. "Hashgraph" names the data structure and the overall algorithm, while "hashgraph consensus mechanism" stresses its role in agreement, alongside alternatives such as proof of stake and practical Byzantine fault tolerance.