Proof of Stake (PoS)
The idea came in 2011 as a solution to all the problems that Proof of Work created. For example, the problem of scalability, transaction times, transaction costs, and electricity costs to operate the network. Ethereum has chosen to leave Proof of Work in favor of Proof of Stake in the future for these reasons.
It means that you become part of the network by investing your money instead of mining. The person who is allowed to validate the next block is produced through an election process. Usually you have more influence the more money you invest, and you are rewarded for each correct validation of blocks that are performed. The selection process is governed by chance, but the size and age of the effort is also taken into account. When you have been rewarded for your validation, age is reset, which ensures that the rich are not always the ones who get the most power.
Should it turn out that the validation is false or an attempt at forgery, one is punished by removing the contribution or part of it. The user can also be banned from the network in the future. In this type of network, blocks are not mined, but are forged.
Normally, all existing coins in a Proof of Stake network are created from the beginning, and the transaction costs go to those who made a bet and "locked" their money for this purpose. Some believe that it is a safer model than Proof of Work as it is cheap and easy to be involved in validating blocks, which means that the number of validating nodes increases. In the case of Bitcoin, for example, it requires huge sums of money to have the opportunity to fight for rewards.
Not all Proof of Stake networks allow anyone to participate and vote. Instead, the people who are allowed to participate are required to have a considerable amount of money to be allowed to participate. The reward is then usually higher than otherwise for those who can afford to join.
Examples of these are Dash.
Proof of Authority (PoA)
Some centralized networks have chosen to whitelist x number of nodes in a network that are approved players. This ensures quality, but also destroys decentralization. It requires that you blindly trust the organization in question.
Examples of these are VeChain.
Distribuerad Proof of Stake (DPoS)
The idea was raised in 2014 and means that you can delegate your efforts and your votes to other nodes. Through a voting process, delegates are elected in a kind of digital democracy.