A blockchain is a record of transactions that many computers hold copies of, where adding to it requires agreement and changing history is impractical.
The structure
Transactions are grouped into blocks. Each block contains a fingerprint of the previous one, computed mathematically from its contents.
Change anything in an old block and its fingerprint changes, which breaks the next block, which breaks every block after. Altering history means redoing all the work since, which is what makes the record durable.
The agreement problem
Many computers, no central authority, and they must agree on what happened. Two main approaches.
Proof of work: adding a block requires solving a computationally expensive puzzle. Doing that costs electricity, so rewriting history costs more than any plausible gain.
Proof of stake: participants lock up currency as collateral and lose it for misbehaviour. Security comes from having something at risk.
What it is genuinely good for
Situations where several parties need a shared record and do not fully trust each other or any single administrator.
Cross-border settlement between institutions. Provenance records across a supply chain with multiple companies. Systems where censorship resistance matters.
What it is not good for
Anything a database would do. If one organisation controls the data, a database is faster, cheaper and simpler. Most blockchain projects that failed were solving a problem that had no trust component.
The trade-off
Everything is slower and more expensive than a database, because many machines do the work instead of one. You pay that cost to remove the need for a trusted administrator. Whether that is worth it depends entirely on whether you have one.