Blockchains are a form of distributed ledger. Each account on the ledger holds a certain balance. For this balance to be moved, it must be reduced from the source account then added to the destination account. The process of checking this maths and updating balances is done by block producers. Each block is a collection of these transactions. Once a valid block is produced, any further transactions are placed on the next block. Previous blocks cannot be edited or deleted so the transactions are permanent.
For an account owner to agree to move funds from the account, they must sign a transaction using the private key associated with the account. This ensures that only the account owner can deduct a balance from the account.
Most blockchains also do more than merely track balances. They contain programs or smart contracts which are just sets of code stored on-chain which can complete interactions as defined in the code. Users interact with the program by signing transactions with instructions. These instructions are usually abstracted away by wallets.
Each blockchain uses different methods for adding transactions to blocks, and blocks to the chain, and checking the blocks are mathematically valid. In all cases, there is a delay between the user submitting the transaction and the transaction being confirmed. This delay can range from 15 minutes with bitcoin to a few milliseconds with Solana. Each valid confirmed transaction gets a transaction hash which can be used to check the state and outcome of the transaction using a block explorer.