Ledgers are fundamental to civilization with the earliest versions of ledgers being developed simultaneously with the development of writing. At its simplest a ledger is just a way of structuring data to make it easier to reach a consensus about the facts behind that data. At a more abstract level, it is possible to conceive organisations simple as ledgers of contracts and capital.
Ledgers are typically used to confirm:
Ledgers are especially useful for helping to determine possession and ownership. Here it is important to note that possession in of itself does not denote ownership. An example of this is a rented car. Whilst at a given moment in time I may be in possession of the car, I am clearly not the owner.
Most ledgers today require some level of coercion in order to be enforced which is typically done by the government/judicial system. However, developers of decentralized ledgers believe it is possible to maintain and secure the ledger through the use of appropriate protocols and incentive structures.