Collective agreement reached among nodes in a network.
Consensus is easy to agree between, say, two parties (Alice invites Bob to her house; Bob agrees.) But as the number of parties (or nodes on a network) increases, consensus becomes more difficult to achieve.
In order to coordinate in their common goal, all the nodes on a network need to agree on a "single source of truth" — that is, they need to be confident that the information they are getting is accurate, even in the event that multiple other nodes on the network fail.
Every node in the network (or every participant in, for example, the Bitcoin blockchain) needs to have an identical copy of the ledger, and there must be agreement that the version of the ledger held by every node is accurate. In order to achieve this, blockchain projects use distributed consensus algorithms.
In the Bitcoin blockchain, consensus and coordination is achieved through Proof-of-Work, as proposed and developed by Satoshi Nakamoto. Other consensus mechanisms exist, including Proof-of-Stake and Delegated Proof-of-Stake.
However, most distributed consensus mechanisms have the same basic properties. First, they are based around a stake — some store of value that a proposer is putting up, such as currency or computing power. They also involve a reward for validation (that is, for mining a new block,) which is generally taken in the form of a coin native to the blockchain in question. They also work on the basis of transparency — that is, other users need to be able to see if a validator or proposer is attempting to cheat or defraud the system.
Proof-of-Work is the foundation for blockchain-based consensus, and remains the basis of the Bitcoin blockchain, on which new blocks are mined. However, there are new consensus mechanisms being used on other blockchain projects. Some of these, such as Proof-of-Stake, are now achieving greater adoption.