Custodian
In short: A custodian is responsible for the safe custody of assets that belong to an organization or to individuals.
A custodian is responsible for the safe custody of assets that belong to an organization or to individuals. In crypto, this means holding and protecting the private keys that control the coins on the owners' behalf.
Why it matters
Whoever controls the private keys controls the funds, so choosing between a custodian and self-custody is one of the first security decisions a crypto user makes. When you deposit coins to a centralized exchange, the exchange becomes their custodian: it holds the keys, and your account shows a balance you can trade or withdraw. Custodians typically keep most funds in cold storage, use hot wallets for day-to-day withdrawals, and require internal approvals, while their clients carry the risk that the custodian fails or restricts access. The saying not your keys, not your coins sums up the self-custody view.
Example
Alice buys some bitcoin on an exchange and leaves it in her account, so the exchange is her custodian. Later she withdraws the coins to a hardware wallet whose seed phrase only she knows. From that moment she is her own custodian: no company can freeze the coins, but no company can help her recover them if she loses the seed phrase.