A non-custodial wallet grants you full, complete, and exclusive ownership and responsibility for your private keys to your blockchain wallets. This type of wallet can come in many forms, such as hardware that looks like a pen drive, a mobile app, or a desktop app. This type of wallet is much less likely to be hacked. Transactions are still signed in the wallet and the confirmation is what gets sent online to the blockchain. There is no trust in a third party required, but you must have a lot of trust in yourself. If you lose your private key there is no way to retrieve it.