Proof of Stake is a different form of algorithm for keeping the security of the blockchain in check – specifically to avoid some of the future potential pitfalls that could occur with Bitcoin such as the Tragedy of the commons.
The tragedy of the commons is where miners are driven out of the network as profitability falls on decreasing transaction fees available.
One of way of circumventing this problem is by including a proof of stake – or where you mine, or mint, in proportion to the number of coins that you hold.
There are some key advantages to this - people are encouraged to hold onto their coins as they essentially earn interest which avoids the tragedy of the commons in Bitcoin with Proof of Work. This is also its blessing and its curse as it causes friction with money flow as seen with traditional money systems of deflationary and inflationary spirals depending on the level that interest is set at.
Another key advantage is that proof of stake is a lot more energy efficient that proof of work as it can reduce the level of complex calculations and guess work involved that is inherent to it. Proof of stake only means people holding the currency earning it by verifying that they own it.
So proof of stake requires less computational work as minting a new block depends on the stake the user has. But this can lead to issues such as blockchain forking as with proof of stake when a fork occurs all minters or miners have an incentive to mint and mine both parts of the branch unlike with Bitcoin and proof of work.