Proof of Stake (PoS) is a category of consensus algorithms for public blockchains that depend on a validator’s economic stake in the network. In other words, miners get greater rewards for holding more coins. The very first PoS coin was Peercoin.
In PoS-based public blockchains (e.g. Ethereum’s upcoming Casper implementation), a set of validators take turns proposing and voting on the next block, and the weight of each validator’s vote depends on the size of its deposit (i.e. stake). Significant advantages of PoS include security, reduced risk of centralization, and energy efficiency.
One key take-away is that the PoS algorithm should be able to avoid the possibility of a 51% attack. This is because even if a miner did build up a 51% stake in a PoS coin, it would be in his best interests not to perform such an attack on a coin that he held such a large stake in.