This was totally confusing to me, but fortunately there are a lot of really great articles about this on the web.
Proof of Work and Proof of Stake both describe algorithms for cryptocurrencies. A cryptocurrency is either a Proof of Work (POW) currency or a Proof of Stake (POS) currency (or token). The most well known Proof of Work currency is Bitcoin.
So a quick explanation of Proof of Work vs Proof of Stake:
Proof of Work uses an algorithm by which individuals compete to the first person to guess the 64 digit-hex key of the next block. This is called “mining”, and when a miner successfully wins this race they earn a bitcoin reward. Once this transaction is verified, it gets added to a block, and this is added into the blockchain and distributed via the public ledger. Basically you are using tons of computer power to run through millions of hash guesses. This computer power uses a lot of energy, and even if you guess the key you aren’t guaranteed the reward. You have to get there first. This is called proof of work.
Individual users with a desktop computer can’t even really compete in the mining game, its really now massive computer farms with heavy duty ASIC machines built for mining. Maybe I will try cloud mining!
Proof of Stake is an entirely different algorithm for selecting who gets to mine the next block on a blockchain. Since PoS Mining uses so much energy, some coins opt for Proof of Stake, which decides who gets to mine the next transaction by virtue of a different set of rules. Each coin is different, but it tends to be by things like who has the most coins (wealth), the length of mining, or even just by random selection.
Part of the reason that some developers are choosing to make their coins Proof of Stake as opposed to Proof of Work is based on the energy usage, and the theoretical economic theory, The Tragedy of the Commons.