Proof of Work (PoW) and Proof of Stake (PoS) are the two protocols having same purpose and the purpose is to establish consensus on the blockchain i.e. we want to agree on a certain state of blockchain or all nodes need to agree on a certain state of blockchain. So guys when a block is produced, it needs to be appended to the blockchain and all the miners try to mine it. Let me clear it.
When a block arrives, we hash all of the transactions in the block and we get a hash of those transactions then we need to find proof or in the blockchain it's called "nonce (random bits)". Again from starting, the miners recieve hash of the new block with transactions and they need to find the other part (Proof or nonce) so when they append those two together into a string and after hashing this string, they should recieve an answer and the miners know what they are looking for and this answer is publicly known.
In bitcoin blockchain, the string of the answer which is a hash, it need to start with certain number of zeroes and all the miners know this thing. The whole challenge is to find the nonce and it's very hard to come up with the correct nonce so the miners try different different nonces with the hash of the block to find the correct nonce, they do see their new hash which start with certain number of zeroes and if they dont find zeroes in starting then they again try a new different nonce. They try this thing again and again to find the correct answer. It takes a lot of time before a miner finds the correct nonce to get the correct answer. The first miner who finds this correct nonce for that block will be awarded some bitcoins and the block will be appended to the blockchain and all of the miners will be notified that this block is mined and appended.
It's costly and inefficient because electricity is being used by millions of machines to find that nonce.
This was the explanation of Proof of Work.....i hope you got it. Lets come to Proof of Stake.
It is like PoW however it tries to eliminate the issues related to CPUs and GPUs. In PoS, the miner who have the most money in wallet will be most likely to mine the new blocks and it will no longer be called mining. Instead of counting CPUs or GPUs or how much money you have spent on hardware, it counts the money you have in blockchain or in your wallet and this is done because the system thinks that when you have a lot money in the blockchain, you are less likely to harm the network and to mine the false blocks and if you mine false blocks you will decrease the value of your money in the blockchain. In this, the person with more money in the blockchain will be able to mine the new block.
Ethereum is trying to switch on PoS with it's new PoS protocol called Casper, it is a feature which states that the malicious validator (in case of PoS, miner is called validator) who validates the block will lose money if he/she validates a malicious block that is not correct.
I think it's enough clear now b/w PoW and PoS. I want to add more. In PoS, when a validator validates an algo or block, he just put some money or stake to validate that block, you cant get it back until the block is validated or you can actually lose your money which is put to solve that block, if you validate a false block in casper update.