A blockchain is a ledger (piece of paper that records transactions) with perfect memory of every single transaction that has ever occurred within a certain network. This makes double spending within the network impossible as you can trace every single coin in the network back to its origin, verifying that your trading partner does, in fact, have the coin he claims to. This removes the need for a trusted third party to oversee/verify transactions.
This is important for banks because currently they pay a 3rd party clearing house to process transactions and transfers between them. They cannot trust each other because their transactions are not linked together in one shared network.
If they all agree to get on a shared blockchain (ledger), they can split the costs of continuously verifying the validity of the blockchain (leder) between them, rather than paying a 3rd party for the same service. This would reduce fees within the industry and reduce time between transfers.
I used to be very excited about blockchain technology and its implications for creating micro/private currency networks. Something about the financial industry's interest has really turned me off the blockchain and its possible implementations.