Custody implies taking full responsibility for something. In the context of cryptocurrency, it means safeguarding the private keys of a wallet. This can be done by either the owner of the wallet or a third party such as a custodian or exchange. Custody is a critical security measure because if someone else has access to your private keys, they can spend your coins.
Besides crypto custody, there is also fiat currency custody. In this context, it’s about storing your fiat currency with a custodian (i.e. having an account on an exchange side) who should provide security and insurance to protect your assets. Only a few crypto platforms currently offer attractive fiat custody services, but that’s changing with time.