Ledger refers to a process of book-keeping in a given monetary system. A ledger keeps track of its users balances and records all transactions that take place in the system. In cryptocurrencies, a ledger is the blockchain itself, which are distributed to all the users instead of being centralized.
Every financial systems will always have ledger that keeps records of every transactions and that is why it is so important to every monetary system. Centralized ledger is even more important as it becomes a crucial point of attack for criminals. Decentralized ledger is more secure as the algorithm ensures and verifies every transactions. However, it is not impenetrable and it has its own security concerns.