The blockchain is Bitcoin's public ledger. It is a list of blocks, and each block contains a batch of transactions plus a small header.
How blocks link
Each block header includes the hash of the previous block's header. A hash is a digital fingerprint: change one character of the input and the output is completely different. Bitcoin uses SHA-256 (applied twice).
Because every block points to the fingerprint of the one before it, changing an old transaction would change that block's hash, which breaks the link to the next block, and the next, all the way to the present. To get away with it, an attacker would have to redo the proof of work for every block since — faster than the rest of the network combined.
What is inside a block?
A block holds transactions that spend UTXOs (unspent outputs from earlier transactions) and create new ones. Waiting transactions sit in the mempool until a miner includes them. A new block is found on average every 10 minutes.
Confirmations
When your transaction lands in a block, it has one confirmation. Every block mined on top adds another. Six confirmations (about an hour on average) is a common rule of thumb for large payments, because reversing that many blocks would be extraordinarily expensive and unlikely to succeed.
Who keeps it?
Anyone. Every full node stores and verifies the whole chain from the genesis block. There is no master copy — the valid chain is simply the one with the most accumulated work that follows the rules.