This is a highly simplified ELI5 attempt to explain the difference between the two best-known mechanisms for consensus finding in distributed systems.
Imagine both systems, proof of work and proof of stake, as democratic elections. But instead of using your ID card (proof of citizenship) to cast your vote, you use something else.
In the case of proof of work, you need to do some work. For example (and in most cases with crypto) you invest computing power. The more you invest, the stronger your voting rights in the system. You can already see the divergence from a democracy — not all votes are equal.
Looking outside of the crypto world for an analogy, you could say gold is proof of work — people need to actually invest work (mining) to get gold. The more work you invest, the more gold you get. But it's not called that, because with gold you don’t have to prove it, you just have to do it. In crypto, you prove you’ve done the work by showing your results, which need to correspond to certain rules (look up difficulty, nonce, etc).
In the case of proof of stake, the vote is no longer tied to the work you put in, but instead, it is tied to your current existing participation (stake) in the network. So, as an analogy, if gold worked via proof of stake (this is just a way to picture it, it doesn’t), whoever owns 1kg of gold will have a thousand times more voting rights than someone who owns 1g of gold.
So now you’re probably asking “ok, but what do I vote for?”. Simply put, you vote for who gets the next block. Even more simply put, you vote for who gets the next mined coins. The more work (or stake) you put into the network, the higher your chances of winning that vote.
Allow me to just add my own, personal, sometimes unpopular opinion here real quick, and say that proof of stake is a the-rich-get-richer scheme and comes with many more troubles and incentive issues than proof of work.
I wrote an actual explanation for PoW here for anyone who wants to dig much deeper.
