Who actually holds the key
A custodial wallet, like an exchange account, means the exchange holds the private key on your behalf. You authenticate to their platform, and they sign transactions for you. This is convenient and shifts key management risk onto them, but it also means you don't actually own the crypto in the cryptographic sense: you own a claim against the custodian, and if they freeze withdrawals, get hacked, or become insolvent, your access depends entirely on their solvency and goodwill, not on cryptography.
A non-custodial wallet means you hold the private key (or seed) directly, and no third party can move funds without your signature. This is what 'self-custody' means. It removes counterparty risk entirely, but it also removes any safety net: there is no support line that can reverse a mistaken transaction or recover a lost key. The security burden shifts fully onto whoever holds the key.