A blockchain is a transaction database shared by all nodes participating in the network. A complete copy of a blockchain contains every transaction ever executed in the network.
Bitcoin is the original blockchain application with decentralized nodes, a decentralized system for maintaining the ledger (based upon miners that compete for newly minted bitcoins and transaction fees), where the values that are transacted within the ledger are called bitcoins.
Traditionally, a publicly used ledger is kept consistent by two methods: 1) the ledger is maintained by a centralized entity, or 2) the ability to transfer values between cells of the ledger is controlled in some manner.
For the public ledger of USD, both approaches are used. Banks and financial service institutions are regulated entities that have the authority to move values between cells in the ledger, and printed money is difficult to duplicate.
If I buy a cup of coffee for $3.00 with my debit card, a number of regulated entities are involved in deducting $3.00 from my cell, adding $2.66 to the merchant's cell, and dividing the remaining $0.34 among the cells of the processors involved. To ensure that none of these entities creates USD in the process (increasing the overall value of the USD ledger), they must file copies of their ledgers with the U.S. government, and are subject to audits. (Of course, loans can increase the overall value of the USD ledger, but this practice is also regulated and audited by the U.S. government.)
If I buy a cup of coffee for $3.00 with printed dollars, my cell's value is first reduced by $3.00, I am given three dollar bills that represent that change in value, I give those bills to the merchant, and then the merchant exchanges those bills at her bank to increase her cell's value by $3.00. To ensure that no bills are created in the process, dollars are made difficult to counterfeit.
In a blockchain like bitcoin, the ledger is public and distributed. Here, the transfer of values between cells of the ledger is kept consistent due to an immutable public history of the ledger, and the inability to transfer value between cells without a validation intrinsic to the originating cell. That is, in a blockchain, I cannot reduce the value of a cell without the numeric key for that cell, and the current value of that cell is reflected in the public history of the ledger.
Because of these differences, a blockchain ledger has the potential to be advantageous over traditional ledgers to the extent that traditional ledgers are less efficient. These inefficiencies can range from the costs of regulating ledger-modifying entities (i.e. the costs involved in auditing banks and transaction services), to the costs of insuring against asset duplication (i.e. title insurance).
Still, unregulated node decentralization (like bitcoin) isn't necessary to take advantage of the benefits of a blockchain ledger. For example, a stock exchange could run on a blockchain ledger, yet the distribution of nodes (and possibly clients) in that blockchain could be controlled by a central authority. One can also imagine a currency whereby nodes were run by government entities, and financial institutions ran registered clients.
Of course, there are costs and risks involved in the maintenance of a blockchain ledger. However, it is likely that blockchain ledgers will ultimately prove to be more advantageous to than traditional public ledgers for the management of some assets.
Whether or not the original blockchain application succeeds, I believe that blockchain ledgers will replace many traditional ledgers in the coming decades.