What is a Bitcoin wallet?
An app or device that holds your keys and makes, signs and sends payments. The bitcoin itself stays on the blockchain.
A wallet doesn't store coins. It stores the keys that can spend them, and works out your balance by adding up the pieces of bitcoin those keys control.
If an exchange holds the keys for you, you're trusting the exchange. With a self-custodySelf-custody (non-custodial)Holding your own keys, so no company can freeze, lose or take your bitcoin. The alternative, a custodial service, holds them for you. wallet, only you hold them.