A bitcoin wallet stores your private key. When sending bitcoin, it creates & signs the transaction, then broadcast the transaction to the network. When receiving bitcoin, it will read the blockchain & notify you that you have received funds.
If you run a full NODE, your NODE will store your private keys (wallet). SPVs also store your private keys (wallet), but no blockchain, so it must query a NODE(s) on the blockchain to execute transactions.
Paper wallets are simply your private keys written on a piece of paper, but because it is offline, you must import the key to an SPV or full NODE to execute transactions.
Hardware wallets (looks like a USB drive or âthumb driveâ) are offline also, so to execute transactions you connect (via USB port) to an online application (i.e. LEDGER). It can sign the transactions and broadcast, but never gives the private keys to the computer.
Hosted wallets, such as wallets on Coinbase, etc., you do not store your private keys on these, as you personally have no private keys for these wallets; technically, it is the Companyâs wallet (private keys). You simply login and your transactions are managed by the company.