If you listen to people talking about cryptocurrencies, the term "noncustodial" comes up fairly often. In legal circles, this is often used to refer to noncustodial parents (i.e. the parent without custody of their children after a divorce). Seems odd that it also relates to money, so let me clarify.
When you hold a dollar bill, you have physical custody of your money. The U.S. government can't stop you from using that money. In order to take away your financial privileges, physical force has to be employed. All physical money is non-custodial.
Most of the world's money today isn't paper. It's bits and bytes on various computers at the world's largest financial companies and government databases. The government (or anyone else with the right access to the money computers) can take away your financial privileges with a keyboard. This is custodial money. You may have legal custody of your money, but a simple accounting error could make it very difficult for you to prove that.
If you want to continue using apps and online services for managing your money, you are giving up physical custody of your cash. If society moves away from cash entirely, you have no choice but to use digital money that someone else has custody over.
Currently, the only solution that enables a digital financial system without giving up individual custody of money is blockchain. If you want noncustodial money that no one else can influence without going through you, cryptocurrency may be for you.
P.S. This applies to more than just money. If you use facebook, dropbox, etc. then your digital identity, online social presence, and personal data is all custodial as well. Blockchain fixes this.