Proof of Work has a few problems:
In Proof of Stake, no electricity is spent solving cryptographic puzzles. Instead, holders of the currency can “stake” their tokens, which nets them some benefits.
When the tokens are staked, they are put into a pool of all currently staked tokens, which then decide on the order of transactions in the blockchain.
In Delegated PoS (DPoS), a fixed number of elected nodes are responsible for creating blocks in a round-robin order. These are voted for by token holders. This is kind of like how democracy works. This fixes the scalability issue of a blockchain, but sacrifices decentralization.
In Bonded PoS (BPoS) there are no elected nodes, but voting power is proportional to the stake. Thus, wealthy participants are better off.
In Pure PoS (PPoS) there are no elected nodes. Instead, a random set of participants are chosen from the staking pool and given a cryptographic puzzle that when solved, validates a transaction on the blockchain. As long as 2/3 of the participants were honest, this change is committed to the blockchain. This makes it so the system is scalable, secure, and decentralized.