A decentralized ledger is a record of all transactions on a network. This ledger is maintained and updated by many independent nodes, who collaborate based on a ruleset established by the protocol. Bitcoin uses a blockchain and a Proof-of-Work mechanism to organize the network and maintain its decentralized ledger.
Traditional banks use centralized ledgers to track balances. Each bank branch periodically updates this central ledger, but this ledger is neither public nor auditable. The Bitcoin protocol changes this paradigm by allowing anyone to read and write directly to the ledger. Anyone is capable of publishing a Bitcoin transaction. Miners will add that transaction to the blockchain, and anyone can query the blockchain to check their balances and transaction history.
All nodes in the Bitcoin network keep and validate identical copies of the ledger so that there exists no central point of failure or fraud. Whereas executives at a traditional bank can conspire to arbitrarily change the ledger at their own bank, no one is capable of dishonestly altering the Bitcoin blockchain. This gives Bitcoin ultimate security and trustworthiness.