A wallet managed by a third-party service.
A hosted wallet in crypto is a wallet managed by a third party, such as a centralized exchange. Other providers of hosted wallets can be funds like Grayscale. Hosted wallets have the benefit of keeping funds secure even if a user forgets their password. With a non-custodial wallet, losing the private key is synonymous with losing access to their cryptocurrencies. With hosted wallets, users can turn to customer support to regain access to their funds.
Hosted wallets are wallets with custodians like centralized crypto exchanges, funds in traditional finance, or other types of middlemen. The custodian hosts the wallet for the customer, similar to how a bank hosts an account. Users retain full ownership but not the custody of their crypto. This makes the wallet more convenient to use but has the downside that funds can be frozen for arbitrary reasons.
MetaMask is not a hosted wallet but a non-custodial one. With a non-custodial wallet, the user retains custody of their funds and cannot be frozen out of their account.
Hosted wallets transfer the security liability to the service provider. This means that the wallet’s host is responsible for keeping the wallet secure. In practice, this refers to hacking risks and the risk of misusing customer funds. For instance, FTX had a popular hosted wallet before the exchange collapsed under a bank run that exposed it had used customer funds for trading. Thus, hosted wallets can be considered risky in their own way.