Proof of Work
A proof of work is a piece of data which is difficult (costly, time-consuming) to produce but easy for others to verify and which satisfies certain requirements. Producing a proof of work can be a random process with low probability so that a lot of trial and error is required on average before a valid proof of work is generated.
Hashcashis proof of work mechanism bitcoin uses. Basically miners are trying to find a hash which is lower than target by changing nounce, and brute force is the only way to find such hash. While on the other hand, verifying the block + nounce actually results to the hash is a very simple process. It only requires one time hash calculation to verify.
Proof of Stake
Proof of stake is a proposed alternative to Proof of Work. Proof of stake, like the name indicates, want to put emphasis on actually owner the stake in the network. Put it simple, you can think it like you have more voting power if you have more stock shares. This solves the problem of actual miner wants to harm the network to destroy the value of the network, because in order to harm the network by dominating the network, the miner actually need to own dominate stake of this network. But if he has dominate stake of the network, he won’t have incentive to harm the network to destroy the value of his stake he owns.
Proof of Burn
Proof of burn is another method for distributed consensus and an alternative to Proof of Work and Proof of Stake. It’s more similar to PoS than PoW. Instead of owning stake in order to have higher chance of mining block, you burn your coin in order to mine. You can think this as you use your money to buy entry of a lottery, the more ticket you buy, the higher possibility to win the lottery.
Proof of Publication
Proof of publication is a method that uses Bitcoin or Bitcoin-like technologies to authenticate that a certain information was published at a certain date, or was known at a certain date. One example of this is https://poex.io/, which basically hashes your document and publish it to blockchain. Then you will have the proof that you have such document on certain date.