The double-spend problem
A digital file can be copied perfectly. That makes digital cash hard in a way physical cash never was, and solving it is the whole invention.
Hand someone a ten-pound note and you no longer have it. The physical object enforces the rule. Send someone a digital file and you still have it, because copying is what computers do.
Why this breaks digital cash
If money is a file, nothing stops you sending the same file to two people. Both recipients see a valid payment. Only one can be honoured, and nothing in the file itself says which.
Banks, card networks and payment processors all solve double-spending by keeping one authoritative ledger and checking each transaction against it. This works extremely well. It also means the operator can refuse a transaction, reverse it, or fail — and that you must trust them not to.
What makes the distributed version hard
Remove the single ledger and you have thousands of copies. Now suppose two conflicting transactions spending the same balance are broadcast at almost the same moment, reaching different parts of the network first.
Every machine must end up agreeing on the same answer, with no coordinator, while some participants may be actively lying, and while messages arrive out of order or not at all. This is a known and genuinely difficult problem in computer science, studied for decades before anyone applied it to money.
The insight
Checking a signature is easy and every node can do it alone. What cannot be done alone is deciding which of two conflicting transactions came first, because "first" has no meaning across a network with no shared clock. Bitcoin's contribution was a method for producing an ordering that everyone converges on, and making rewriting that order expensive rather than impossible. See proof of work.
Why it is expensive rather than impossible
This is the part most summaries skip. Nothing prevents someone rewriting recent history if they command enough of the network's capacity. The design makes doing so cost more than it is likely to yield, and makes older history progressively more expensive to rewrite than newer history.
So a confirmed transaction is not permanent in the mathematical sense. It is permanent in the economic sense, which is a weaker and more honest claim, and it is why exchanges wait for several confirmations before crediting a deposit. See consensus and finality.