Blockchains are basically a record-keeping system that can store a lot of digital information securely, in a relatively small space, and keep it organized in a step-by-step way.
Every time a transaction happens in a system using a blockchain, it is logged. It is also linked to the transaction that happened before it, creating a long "chain" of connected information. This means that every single transaction can be accounted for, and new transactions can by verified by checking their information against the entire history of the system. The interconnectedness of this model of record keeping makes it highly secure and enables transparency.