A custodial account is an account one party holds and administers for the benefit of another. That definition is broad enough to be almost useless on its own, which is the point of this page: the phrase is used for three arrangements that have very little to do with each other, and the answer to nearly every practical question turns on which one someone means.
The minor's account. An adult opens an account in their own name as custodian for a named child, under the state's enactment of the Uniform Transfers to Minors Act or its older predecessor. The transfer is a completed, irrevocable gift, the property belongs to the child, and the custodian's job is to manage it and hand it over when the custodianship ends. This is what most people mean.
The tax-code account. Several parts of the Internal Revenue Code define a savings vehicle as a trust and then add a provision saying that a custodial account counts as one. IRC 408(h) does this for individual retirement arrangements, providing that a custodial account "shall be treated as a trust" where the assets are held by a bank or an approved person and the account would otherwise qualify, and that the custodian "shall be treated as the trustee thereof." Section 403(b)(7) does the same job for a category of 403(b) accounts, and section 530 for Coverdell education savings accounts. Here the word "custodial" is a statutory labeling device rather than a description of who owns anything: the account owner is the individual whose account it is.
The institutional usage. In investing, the word describes the firm that holds securities, settles trades, and produces statements, and "custodial account" is sometimes used loosely for any account at such a firm. That usage says nothing about ownership at all.
A related term worth keeping separate: the custodian in the institutional sense is a company, while the custodian of a minor's account is a person. The two roles share a name and almost nothing else.