Bitcoin Core (BTC) and Bitcoin Cash (BCH) are dependent on something known as "the blockchain". This underlies and structures the Bitcoin system. The blockchain is the vertebrae of the protocol and the glue that holds the network together. It is simply a vast, distributed public ledger of account. It keeps track of every transaction ever made in the network, and all transactions are timestamped and verified by network miners.
This is how it works: miners with specialized computers compete to solve mathematical puzzles with other computers, and once they solve a puzzle they are awarded with some Bitcoin Core, but they also add a “block” of completed transactions to the blockchain for future viewing and verifiability. Once a block is added to the chain the cycle repeats itself, and the computers continue to compete to solve these difficult problems. Every transaction on the blockchain is completely transparent and accounted for in its log. Anyone can see the public keys of any transaction they want (although there are no names associated with transactions). One could go all the way back and view the very first transactions ever made on the first block ever created.
This block was called, "The Genesis Block". (Update: Bitcoin Core is less usable as money due to much higher fees and delayed transaction times. The Core team has also expressed an interest in keeping these fees high since they view BTC as a "store-of-value" and not something to be transacted on a daily basis. In contrast, Bitcoin Cash's transaction fees cost pennies and payments can be validated even with zero confirmations. These facts make BCH the ideal cryptocurrency for sending and receiving money anywhere in the world.)
Double-Spending is the act of using the same BTC twice. There is only a 21 million set cap on the protocol and no more can be produced. So the network protects against double spend by the verification of each recorded transaction. The blockchains ledger ensures that the transactions are finalized by its inputs confirmed by miners. The confirmations make each unique Bitcoin Core and its subsequent transactions legitimate. If one tried to duplicate a transaction the original blocks deterministic functions would change showing the network that it is counterfeit and thus the transaction would not to be accepted.
Banks and accounting systems use ledgers to track and timestamp transactions. The difference is that the blockchain is completely decentralized and open source. This means that people do not have to rely on or trust the central bank to keep track of the transactions. The peer-to-peer blockchain technology can keep track of all the transactions without the fear of having them erased or lost. Furthermore, the blockchain, because of its open source nature, is more versatile and programmable than central banking ledgers. If programmers need new functionality on the blockchain, they can simply innovate on top of already existing software through consensus. This is difficult for central banks because of all of their regulations and central points of failure.