A custodial wallet is a type of wallet whose private keys are secured by an entity other than the asset's owner and is the opposite of a self-custodial wallet.
Custodial wallets come with various tradeoffs from self-custodial wallets, which can vary widely depending on the custodian's service agreement and operational performance. These tradeoffs, some of which may serve both as features and/or bugs depending on a specific user's needs, include but are not limited to: supported assets, supported swaps, transaction fees, two-factor authentication, KYC/AML requirements, supported jurisdictions, whitelisting, API support, withdrawal speed, withdrawal fees, proof of reserves, audits, insurance, applicable regulations, and more.
A custodial wallet exists whenever one entity transfers possession, but not ownership rights, of cryptoassets to another entity for impermanent safekeeping. Fundamentally, it is a transfer of responsibility, and thus liability (but not necessarily legal), and can be done either as part of a personal agreement or as part of a business agreement.