A blockchain is a list of batches. Each batch — a block — holds a set of transactions, a timestamp, and a cryptographic hash of the block before it. A hash is a fixed-length fingerprint of an input: change one character of the input and the fingerprint changes completely, and there is no practical way to work backwards from a fingerprint to the thing that produced it.
Chaining the fingerprints is what makes the history tamper-evident. Because block 900,001 contains the fingerprint of block 900,000, editing anything in block 900,000 changes its fingerprint, which invalidates the reference held in 900,001, which changes 900,001's own fingerprint, and so on to the tip of the chain. Rewriting one old record therefore means rewriting every record since — and doing it faster than the rest of the network is extending the honest chain. That is the whole security argument, and it is an economic argument rather than a mathematical one.
The second problem is agreement: not which history is valid, but which of several valid histories everyone should treat as the real one. This is what a consensus mechanism decides. Proof of work makes proposing a block expensive in electricity, so that competing with the network costs more than the attack is likely to earn. Proof of stake makes it expensive in capital, by requiring proposers to post collateral that the protocol can destroy if they sign contradictory blocks. Different costs, same purpose: make dishonesty more expensive than honesty.
What follows from the design is a set of properties that are genuinely unusual. Anyone can verify the whole history independently. Nobody can quietly delete a record. Balances can be checked without asking permission. What does not follow is any guarantee about the truth of what was written. A blockchain will record, permanently and verifiably, that a company transferred a token representing a shipping container — and it has no way to know whether the container exists. Data that enters from the outside world is exactly as reliable as whoever entered it.
This is the distinction that most enterprise blockchain projects run aground on. If a single organisation controls who may write, a database is faster, cheaper and easier to fix. The chain earns its overhead only where the participants genuinely do not trust one another and there is no acceptable party to appoint as keeper. That is a narrower set of situations than the last decade of announcements suggested, and it is the set where public blockchains have in fact stayed.
One more property matters for reading anything else on this site: blocks arrive at intervals, so settlement is probabilistic rather than instant. A transaction one block deep is far more likely to be reversed than one twenty blocks deep, which is why exchanges wait for a number of confirmations before crediting a deposit. Finality is something a chain approaches, and different designs approach it at very different speeds.