Custody is traditionally used in relation to cryptocurrency wallets, custody refers to how you store your cryptocurrency. In this case, custodial is when you keep your assets in a third party, like a crypto exchange. On the contrary, non-custodial is when you store your assets in a decentralized manner, using digital wallet like MetaMask or a ledger.
Cryptocurrency exchanges allow users to store assets on the platform with a caveat that all trades and transactions are recorded. Some crypto exchanges also have the right to freeze your funds and prevent withdrawal.
It is always recommended that you keep your crypto assets privately as it is more secure.