Proof of Stake
Proof of Stake is a propose alternative mechanism to Proof of Work as a way to mine Bitcoin and sign transactions. It was probablly first proposed here by Quantum Mechanic. With Proof of Work, the probability of mining a block depends on the work done by the miner (e.g. CPU/GPU cycles spent checking hashes). With Proof of Stake, the resource that's compared is the amount of Bitcoin a miner holds - someone holding 1% of the Bitcoin can mine 1% of the "Proof of Stake blocks".
Some argue that methods based on Proof of Work alone might lead to a low network security due to Tragedy of the Commons, and Proof of Stake is one way of changing the miner's incentives in favor of higher network security.
Here is one attempt to describe an implementation of Proof of Stake.
Motivation For Proof of Stake
A proof-of-stake system would provide increased protection from a malicious attack on the network. Secondly, when block rewards are produced through txn fees, a proof of stake system would result in lower equilibrium txn fees. Lower long-run fees would increase the competiveness of bitcoin relative to alternative payments systems.
The Problem of Monopoly
If a single entity (hereafter a monopolist) took control of the majority of txn verification resources, he could use these resources to impose conditions on the rest of the network. Potentially, the monopolist could choose to do this in malicious ways, such as double spending or denying service. If the monopolist chose a malicious strategy and maintained his control for a long period, confidence in bitcoin would be undermined and bitcoin purchasing power would collapse. Alternatively, the monopolist could choose to act benovolently. A benevolent monopolist would exclude all other txn verifiers from fee collection and currency generation, but would not try to exploit currency holders in any way. In order to maintain a good reputation, he would refrain from double spends and maintain service provision. In this case, confidence in bitcoin could be maintained under monoploly since all of its basic functionality would not be affected.
Both benevolent and malevolent monopoly are potentially profitable, so there are strong reasons for believing that an entrepreneurial miner will attempt to become a monopolist at some point. Due to the Tragedy of the Commons effect, attempts at monopoly become increasingly likely over time.
How Proof of Stake Addresses Monopoly Problems
Monopoly is still possible under proof-of-stake. However, proof-of-stake would be more secure against malicious attacks for two reasons.
Firstly, proof-of-stake makes establishing a verification monopoly more difficult. At the time of writing, an entrepreneur could achieve monopoly over proof-of-work by investing about 10 million USD in computing hardware. If price remained constant in the face of extremely large purchases, such an entrepreneur would need to invest about twice in USD to obtain monopoly under proof-of-stake. Since such a large purchase would dramatically increase bitcoin price, the entrepreneur would actually need to invest several times this amount. Thus, even now proof-of-stake monopoly is several-fold more costly to achieve. Over time the comparison of monopoly costs will become more and more dramatic. The ratio of bitcoin's mining rewards to market value is programmed to decline exponentially. As this happens, proof-of-work monopoly will become easier and easier to obtain, whereas obtaining proof-of-stake monopoly will become progressively more difficult.
Secondly, and perhaps more importantly, a proof-of-stake monopolist is more likely to behave benevolently. In a benevolent monopoly, the currency txn continue as usual, but the monopolist earns all txn fees and coin generations. Other txn verifiers are shut out of the system, however. Since mining is not source of demand for bitcoin, bitcoin would likely retain most of its value in the event of a benevolent attack. Earnings from a benevolent attack are similar regardless of whether the attack occurs under proof-of-stake or proof-of-work. In a malicious attack, the attacker has some outside opportunity which allows profit from bitcoin's destruction (simple double-spends are not a plausible motivation; ownership of a competing payment platform is). At the same time, the attacker faces costs related to losses on bitcoin-specific investments which are necessary for the attack. I assume that a malicious attack causes the purchasing power of bitcoin to fall to zero. Under such an attack, the proof-of-stake monopolist will lose his entire investment. By contrast, a malicious proof-of-work monopolist will be able to recover much of their investment through resale. Recall also, that the necessary proof-of-work investment is much smaller than the proof-of-stake investment. Thus, the costs of a malicious attack are several-fold lower under proof-of-work. The low costs associated with malicious attack make a malicious attack more likely to occur.
Why Proof of Stake Would Decrease Long-run Txn Fees To be added (Basic idea is that opportunity cost (i.e. marginal cost) of txn verification is much lower when verification doesn't require excessive use of electricity. Therefore under both competitive or monopoly pricing, equilibrium txn fees are much lower.