Proof-of-Stake is a method for validating a holding of a cryptocurrency. In a normal currency, paper money or coins are evidence of ownership, but in a cryptocurrency, like bitcoin, ledger entries is the only way that ownership can be known. The overall ledger that shows ownership over time is called a "blockchain", and dfferent systems have evloved to ensure that such blockchains have not been forged or altered. The two main methods are proof-of-work and proof-of-stake.
Proof-of-work repeatedly encrypts the records in the blockchain to the point that there is not sufficient computer power to forge or alter its contents. Proof-of-stake allows additional transactions to be added to the blockchain if the payor can prove that he owns the currency in question. As a result, a proof-of-work currency relies upon the computing power of "miners" who are rewarded for their effort with shares of the cryptocurrency while a proof-of-stake currency puts the computing burden on the holders of the currency.