Non-custodial

Custodial

Custodial usually means that someone else is holding the private keys to your funds. A third-party is holding your funds for you. Web wallets are usually custodial wallets where the web wallet operator holds the private keys and you as user, client or customer just hold the login to the web site. In a custodial set-up you do need to trust someone else, because someone else is holding the keys on your behalf.

Non-custodial

Non-custodial is the opposite of custodial. Non-custodial usually means that you (and only you) as a user, customer or client own the private keys to your funds. E.g. the Lightning wallet Eclair Mobile is a non-custodial wallet where the private keys reside on your cell phone, and as such you own your private keys. In a non-custodial set-up you do not need to trust someone else, because you are holding your keys yourself.

Lightning

The Lightning ecosphere has custodial as well as non-custodial wallets. Even cell phone wallets can be custodial. Also Lightning nodes can be custodial if you are using someone elses node, or the node can be non-custodial if you run it yourself.