Custodian
Any party that holds crypto, or the keys to it, on someone else's behalf, such as an exchange holding customer balances.
A custodian is whoever holds an asset for its owner. In crypto that means holding the private key, because whoever holds the key controls the coins. Keep a balance on an exchange and the exchange is your custodian: on the blockchain the coins sit in wallets it controls, and what you own is an entry in its database.
Custody is a spectrum of trust. A good custodian offers convenience (no seed phrases to guard, passwords resettable, support desks that answer) and takes on professional security. The cost is that your access depends on the custodian's solvency, honesty and uptime. The failure of FTX in November 2022 made the point at scale: customers held claims on a company, not coins on a chain, and joined an insolvency queue.
The alternative is self-custody, where you hold the keys and nobody stands between you and the coins, along with nobody standing between you and your mistakes.
The word matters in reading platform claims too. "Cold storage" on a custodial platform means the custodian's keys are offline, which protects against hackers but not against the custodian itself. That distinction is what proof-of-reserves reporting tries, partially, to address.