Proof of Work
Computers solve a mathematical problem as "proof of work" to verify transactions on a blockchain. New transactions are grouped into a block. Miners, aka the computers verifying all the transactions, race to crack the block and recieve payment in the native cryptocurrency to which they are computing. The block is cracked through solving intense mathmatical problems.
This process require a great amount of energy and computing power. Generally miners set up shop in locations where cost of energy is low. This increases their margins.
Proof of Stake
This process is much more energy efficient and becoming adopted quicker. The best way to think of this is like bank collateral that earns high interest. A owner of a cryptocurrency can stake their coins in a communal holdings and earn interest due to the transaction verifications. The more tokens staked the more verifications are done on your accounts behalf therefore making more return.