Blockchains are a specific type of database made up of blocks of data, concatenated together to form a chain.
Specifically, new data comes in piece by piece in the form of transactions. Each transaction is added to a block. Each blocks is chained to the previous block, creating a chronological ordering of data.
While blockchains can store any type of data, the most common use is to produce a ledger: an append-only list of transactions. For example, Bitcion's blockchain stores all the transactions from one wallet (account) to every other. The balance of each account is a reduce over these transactions.
Blockchains use cryptography and distributed computing algorithms to provide immutability, meaning that data entered cannot be changed after the fact. For Bitcoin, this means that transactions are permanently recorded and viewable to anyone.