How does Bitcoin stop anyone from spending the same coin twice, with no bank in charge?

By making everyone keep the same public record, and making that record ruinously expensive to rewrite. Double-spending means paying with the same money twice. Usually a bank or payment company prevents it by checking whether a coin has already been spent, but that makes it a single point of failure. Bitcoin removes the middleman: every computer in the network keeps its own copy of the ledger, the blockchain, and checks every payment against it.

The catch is that messages reach computers at slightly different times, so two payments of the same coin could each look valid to different parts of the network. Bitcoin settles which one counts by bundling payments into blocks, and making each block costly to produce. A miner must find a number that gives the block's hash, its digital fingerprint, a value below a target, which takes an enormous number of tries but only one check to verify. Each block also contains the fingerprint of the block before it, so the blocks form a chain.

When there is a disagreement, computers follow the longest chain, the one that took the most work. To undo a payment, a cheater would have to redo the work for that block and every block after it, and outpace everyone else. The odds of a slower attacker catching up shrink exponentially with every new block, which is why merchants wait for a few confirmations.

The guarantee has limits. Anyone with more than half the network's computing power could rewrite recent history, and smaller copycat currencies have lost millions this way.