A 'consensus algorithm' is a mechanism for verifying transactions in a distributed network. A transaction doesn't take place until everyone (or a large enough subset of everyone) agrees that it can take place (ie., it isn't contradicted by some previous transaction). But how do you get to have a say in these consensus networks? That's what the algorithms decide. The first is 'proof of work', such as solving equations to 'mine' bitcoins. Another is 'proof of stake' in which you 'bet' on the validity of transactions. A third is 'delegated' proof of stake, in which you elect a subset of you to become validators. Related is 'proof of authority', where transactions are validated by approved accounts. The 'proof of weight' algorithms award authority to the largest entities. Then the algorithms get messy: there's Byzantine Fault tolerance, spaghetti algorithms, and more.
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