A wallet or service is custodial if the user is not in control of their own private keys. For example, most exchanges and brokerages are custodial: they store the bitcoin and control the keys while showing the user their balance based on the company’s internal ledger.
Custodial wallets may have extremely high security, but they are also subject to government regulation and cannot be audited. A custodial service’s solvency or security practices cannot be proven by the average user. Likewise, if a user wants to send or receive bitcoin using a custodial service, they risk being censored or having their funds confiscated. For this reason, bitcoiners encourage one another to withdraw their bitcoin to a non-custodial (AKA self-custodial) solution.