Blockchain technology is about achieving consensus for storing data even in the presence of adversarial participants. You can seriously stop here to appreciate blockchains, because the important properties are that it stores data in a way that cannot be tampered with by any single party. Whatever invariants are baked into the consensus algorithm can be trusted to hold by the network.
The original flavor of consensus, proof of work consensus, relies on a particular class of problems which are prohibitively expensive to solve (where the difficulty can be chosen to be arbitrarily hard) but trivial to verify.
The data to store is gathered into a "block" along with the solution to the chosen problem which is agreed upon by the protocol. This problem will involve the previously accepted block, so that these blocks form a chain. Each block in the chain contains data as well as the solution to a problem that refers to the previous block.
These are propagated to all participants, and the protocol dictates that the longest chain is the valid chain. Because the solutions are easy to verify, the network can agree on whether a given chain of blocks is valid. The difficulty is set so that the solution can be produced every set amount of time (10 minutes, say) which means adding onto the longest chain takes awhile to compute. This makes it difficult for multiple chains to develop because for two competing chains, one will fall behind eventually, and then get rejected by the protocol.
Note that with enough computational power, an adversary would be able to generate blocks faster than the rest of the network, allowing them to decide what a valid block is, and this is referred to as the 51% attack. But for a long running chain with many diverse participants and total computational power, this state is seen as infeasible to achieve for an attacker.
Other blockchain variants tweak the consensus algorithm that decides who can add the next block, but I'll leave that out of this answer.
The main take-away: Blockchains are about establishing a protocol for who can commit the next set of data, in a way that prevents a single party from corrupting the data. The chain of blocks in the name come directly from the proof of work consensus algorithm .