IN PLAIN ENGLISH:
If I send money electronically to my friend, John, he can't claim that he didn't receive the money because proof of the transaction was recorded on thousands of computers around the world that can prove that the transaction occurred.
The technical explanation:
Blockchain is the technology underlying all cryptocurrencies. After a set period of time (10 mins in the case of Bitcoin) a new block is created that contains a record of all transactions that occurred during that period. This block is then attached to all previously created blocks, which are linked together in a chain.
This chain, and all the blocks it contains, is an immutable history of every transaction that has ever taken place with the given currency. The fact that this history, also called a ledger, is distributed among thousands of computers that constantly compare themselves against each other for accuracy, makes it an extreme secure system.
This is the beauty of Blockchain technology. It is distributed, transparent and so secure that you can confidently conduct transactions with strangers without the need for a trusted intermediary.