It's solving the "Byzantine Generals" problem ( https://research.microsoft.com... ), which in simple terms is how to reach consensus without trust. In the context of a payment network like Bitcoin, the consensus to be reached is which transactions have occurred, and therefore what account balance each user has. Prior to the Bitcoin network, the only known method was a trusted third party, such as a bank, who keeps a central ledger of transactions and balances. The problem with a middleman is they can extract excessive fees, or arbitrarily decide not do business with you.
Bitcoin solves the problem using a distributed public ledger, with a "proof of work" function and chained hashes for blocks of transactions. The distributed ledger means everyone has a copy, and can independently verify the history of transactions. The proof of work generates consensus by adopting the longest chain of blocks as the true history. It had the most work put into generating it. Chained hashes use the hash of the previous block as part of the data for the next block (along with new transactions). So any change to past data is detectable.
Since the software is open source, and anyone can create private keys and associated account addresses for themselves, nobody can tell you they won't open a checking account or give you a credit card. Finding block hashes and collecting the rewards and transaction fees is competitive, so fees are set by the market, and not by an oligarchic entity.
The Block Chain technology creates trusted records for transaction data in an untrustworthy environment. But since you can hash any kind of data whatsoever, not just financial transactions, you can keep trusted records for any kind of data. This has usefulness far beyond bitcoin itself, although few of those uses have been developed yet.