A hosted crypto wallet is a digital wallet where people are able to store their private keys. In return, the wallet takes care of backing up and protecting their money.
A hosted wallet is more like a standard banking software than other crypto wallets in that you can always recover access to your funds, even if you forget your password or mnemonic phrase or misplace your phone. This option is not offered by other wallet kinds. You will lose your money if you misplace your login information or a device with the wallet installed on it.