The double-spending problem is one of the major issues that every digital currency faces when it is transferred between peers over a distributed network. If a malicious user is able to access the network he could technically send the same unit more than once, which would result in a double-spending of this unit.
A system is needed that can prevent this problem. In the traditional financial system banks solve this problem by being the central trusted entity controlling all transactions. This way they can ensure that this problem does not happen. In order for cryptocurrencies to provide the same security they must be designed in a way that double-spending is prevented and every transaction can only happen once.
Because these networks want to work without trusting third parties, they need a network which creates trust by itself. This is the reason they use blockchain technology which provides a distributed and public ledger. It shares all records of transactions made with cryptocurrencies across the entire distributed network of nodes.
Every attempt at double-spending is therefore registered quickly. This secures the network and allows its participant to rely on it without needing to trust a single entity.