A self-custodial wallet, as opposed to a custodial wallet, depends on the entity that owns the wallet's cryptoassets to bear sole responsibility for safeguarding its private keys.
Self-custodial wallets come with various tradeoffs from custodial wallets, which can also vary depending on the individual custodian's private key storage methodology (such as a paper wallet). In addition to bearing responsibility for cryptoasset security, self-custodial wallets also limit immediately available cryptoasset services to on-chain applications such as DeFi. To access centralized cryptoasset services like those from centralized exchanges (CEXs), cryptoassets must be sent to custodial wallets.
There are many security considerations for self-custodial wallets. They can be either a hardware wallet or software wallet, as well as either in cold storage or a hot wallet. Generally, when assessing combinations of these storage methodologies for self-custody, users must choose to optimize between convenience and security, though many practices can achieve desirable balances of both.