Proof Of Work is an algorithm that is also used by most cryptocurrencies to achieve decentralization between different nodes in the process of adding a specific block to the blockchain.
Miners solve extremely complex and computationally difficult mathematical problems in order to add blocks to the blockchain and confirm transactions.
Each mined block brings a certain reward, each confirmed transaction brings a revenue through the "fee" (the transaction ends in the block but this block has a limited space, passed this it ends up in another block chained to the previous one through the hash). This process requires time and energy (high computational resources) but the revenues derive precisely from: rewards for mined block and fee.  

Which cryptocurrencies use this system? Bitcoin, Litecoin, Monero, Bitcoin Cash, Bitcoin SV, Ethereum 1.0 (although they will probably change Blockchain, moving to Proof Of Stake in 2020), etc  

In Proof Of Stake, the larger the participation ("Stake"), ie the amount of tokens owned by a user, the more likely it is that the system is not being violated.
The blocks of the Proof Of Stake, unlike the blocks of the Proof Of Work, are not extracted, but coined. Participants who have significant participation in the Proof Of Stake systems are selected on a pseudorandom basis to mint the blocks and add them to the blockchain.
But how can such a system work if you don't use energy and supercomputers?
The participants in the network (with many odds), will put their "Stake" into play.
To simplify the concept: whoever has the most Stakes (cryptocurrencies) has the greatest voting power.
The Proof Of Stake is generally applied to pre-mined cryptocurrencies, so as to allow the user to access it through participation.
The only incentive for miners in this system is represented by the transaction fees associated with the specific block coined.  

Which cryptocurrencies use this system? Dash, Tezos, Electroneum, PIVX, Neo, Reddcoin, Qtum, maybe Ethereum 2.0