In the simplest sense, most cryptocurrency wallets are software programs that interface with various blockchain and store your public and private keys. Users can monitor their balance, send money, receive money and conduct other operations. Hardware wallets and paper wallets can be used as well, depending on the users’ preference.
When a person sends bitcoins or any other type of digital currency, they are essentially signing off ownership of the coins to your wallet’s address. To spend those coins and unlock the funds, the private key stored in the wallet must match the public address the currency is assigned to. If those public and private keys match, the balance in your digital wallet will increase, and the senders will decrease accordingly. There is no actual exchange of real currency. The transaction is just that, a transaction recorded on the blockchain.