Bitcoin and most cryptocurrencies expose your entire payment history to the public. Zcash is the first open, permissionless cryptocurrency that can fully protect the privacy of transactions using zero-knowledge cryptography.
Zcash encrypts the contents of shielded transactions. Since the payment information is encrypted, the protocol uses a novel cryptographic method to verify their validity.
Zcash uses a zero-knowledge proof construction called a zk-SNARK, developed by a team of experienced cryptographers based on recent breakthroughs in cryptography. These constructions allow the network to maintain a secure ledger of balances without disclosing the parties or amounts involved. Instead of publicly demonstrating spend-authority and transaction values, the transaction metadata is encrypted and zk-SNARKs are used to prove that nobody is cheating or stealing.
The official Zcash website with a lot of relevant information.