Before we get into what a is, we need to understand what a ledger is. A ledger in computer science is software that stores transactions. A database is different from a ledger such that in a database we can add, remove, and modify records, whereas in a ledger we can only append but not delete or modify.
A blockchain is basically a data structure to implement a decentralized ledger. A blockchain is a chain of blocks connected to each other. Every block contains a list of transactions and certain other metadata, such as when it was created, which is it's previous block, the block number, who is the creator of the block, and so on. Every block maintains a hash of the previous block, therefore creating a chain of blocks linked with each other. Every node in the network should hold the complete copy of the blockchain and, when a new node comes in, it will request and download the blockchain from other nodes.
The major advantages of using a blockchain is that it enables the facilitation of transactions without a central trusted party; data is secured using cryptography, and data is immutable, as blockchain removes friction and reduces risk so settlements happen in real time, and so on. Basically, it automates auditing, makes the application transparent, and provides a single source of truth.
In the real world, private blockchains are used in trade finance, cross-border payments, digital identity, the clearing and settlement of tokenized and digital assets, provenance of ownership of a product, record keeping for critical data, signing contracts, multi-party aggregation (namely, they can be used as a shared master repository for common industry information, allowing members to query for data), payment-versus-payment or payment-versus-delivery, and so on.
Every blockchain node maintains a database that contains the blockchain's state. The state contains the final result of running all the transactions in the blockchain. For example, in a blockchain, the state represents the final balances of all addresses. So when you query the blockchain node for an addresses balance, it doesn't have to go through all transactions and calculate the final balance of the address; instead, it directly fetches the balance from the state of the blockchain. Bitcoin uses LevelDB to maintain the state of the blockchain. Even if the database gets corrupted, the database can be restored by simply running all the transactions in the blockchain.