Proof of work – Miners solve a complex mathematical problem. The miner that solves the problem first adds the block to the blockchain. The network rewards the miner for doing so.
Proof of stake – Requires ownership, or stake, in the cryptocurrency. The more tokens you own, the more block creation power you have. Benefits: eliminates the need for expensive mining rigs, runs on a tiny fraction of the power, and it requires block producers to have a stake in the network.
Delegated proof of stake – Block-creating accounts, called witnesses, are collectively approved by Steem stakeholders. Instead of relying on proof of work to find blocks, the Steem network actively schedules these accounts to improve the time between blocks to 3 seconds.