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Blockchain or cryptocurrency wallets are software programs that store your public keys (your address, a long string of 34 letters and numbers) and your private key (similar in purpose to a password, a long string of 64 letters and numbers) so users can send and receive and monitor their balance.
When a person sends you bitcoins for example, their wallet signs off ownership of the coins to your wallet. The transaction is completed when the blockchain verifies that your address matches the private key in your wallet. If you created a wallet and faked an address you would not be able to unlock the coins because the blockchain would need the private key to do so.
It’s important to be aware that no actual coins are exchanged between the sender and receiver. The wallet holds an address and keeps a record of all the transactions and therefore the balance.
So in summary, the blockchain keeps a record of every transaction that has been made, and a blockchain wallet records the transactions that a particular address made.