I will dive deep into this subject to bring some clarity about these two mechanisms used in the blockchain. They're pretty different from each other but they're the core of what we're all using today.
To start we need to make some introductory subjects in order for you to understand what are talking about.
The database structures for transactions of cryptocurrencies is the blockchain, which stores the entire transaction history. The name BLOCKCHAIN origins from the fact that all transactions are bundled into blocks; each block in the blockchain references a previous block. Each node that is part of the blockchain, has a copy of it in itself, which is synchronized with other nodes using a user to user protocol.
Any implementation of digital currency must have a way to secure the blockchain against attacks from hackers:
In Bitcoin, you have what is called Proof Of Work, an algorithm in the form of block mining. Each node that wants to mine is required to solve a very difficult mathematical problem to ensure the validity of the mined block and the solutions are rewarded with satoshis of Bitcoin.
Why you ask this is secure? Because the attacker will have to overcome all that mathematical problems to try and break the network. This is virtually impossible since it will require two main things which are pretty difficult to get:
An obscene amount of computational power to process and solve the algorhytms
Electricity that's gonna be spent powering the PC.
This makes POW expensive, inneficient if you wanna refer to it that way, but a lot more secure. This is exactly the problem that POW it's facing. Its expensive and inefficient.
Here it's where Proof of Stake comes to mind (to solve this problem that PoW it's having)
Proof of Stake algorithms comes as another possible decentralized ledger implementation with security but NOT BASED ON EXPENSIVE COMPUTATIONS.
The idea behind this implementation it's pretty simple: users with the highest stakes in the system have the most interest to maintain a secure network, as they will suffer the most if the reputation and price of the cryptocurrency would diminish because of the attacks. To mount a successful attack, an outside attacker would need to acquire most of the currency, which would be prohibitively expensive for a popular system. This is by no means profitable which will deter attackers by a long shot.
So to sum Proof of Stake up think about it like these: instead of using big rigs for mining power and consuming a lot of electricity, the probability of creating a new block in the blockchain and receive the reward it's proportional to the stake a user owns in the system. As explained above, it's pretty simple but a little less secure.
Hope you've enjoyed this introduction to PoW and PoS.
Thanks for the read :)