A zero-knowledge proof is a cryptographic method to prove that you know something without giving any other details about it, except that the fact that you know it. For example, zero-knowledge proofs could be used to show you have assets of more than a million dollars, without showing the exact amount you own or the transactions that make it up.
Zero-knowledge proofs are an ideal fit for providing privacy on blockchains. On a public blockchain, everyone has to be able to verify the authenticity of transactions and so every transaction is posted for everyone to see along with its entire history.
Zero-knowledge proofs allow others to verify that a valid transaction has happened without giving any other information thus providing privacy while retaining verifiability.
Zcoin uses zero-knowledge proofs in its privacy mechanism to allow people to burn their coins and redeem them later for brand new ones with no previous transaction history without showing which coins were burnt.