Let's start off by taking a look at the dictionary definitions of what Custodial is & then look at how it tracks within the conversation of Bitcoin, Ownership & Privacy.
The Mirriam Webster definition of Custodial is:
immediate charge and control (as over a ward or a suspect) exercised by a person or an authority.
The Cambridge Dictionary definition of Custodial is:
referring to or having the responsibility of protecting, caring for, or maintaining something or someone.
In short, a Custodial Account is something you have access to but is maintained & owned by another entity, whereas a Self-Custodial (Non-Custodial) Account is something only you have full access to & ownership over. Let’s use part of my example above to keep with the story of the bank to further our understanding of Custodial Accounts/Services.
As stated before, when you open a bank account, you are essentially opening an account with the bank that they have full control & maintenance over & which they give you access to. This is seen as a custodial relationship because the Bank affords you “easy” access to your funds when you need them & they charge you fees for these services, including if you're not transacting but merely storing your funds for the long-term. Now, the kicker with custodial services like a bank is that they own the account & afford you access to it.
This means they can process payments for you & assist in making sure those payments actually reach their intended destination, but they can also stop you from making that very same payment while still charging you for your attempt, as this is a service they provide.
It doesn’t stop there, just look at what happened in Canada & China when Citizens tried to withdraw all their funds from their bank branches. They were blocked from making those withdrawals entirely, effectively freezing those client’s bank accounts at the whim of the bank. It should be noted that this action is completely legal, as the account belongs to them & not to the customers