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.