Custody in finance is covered in full in the entry Custody Service, which explains what custody providers do, how the service is priced and how it is regulated. This page gives a short definition and explains the general term.
Custody is the holding and safeguarding of financial assets by one party on behalf of another, together with the administration that goes with it. For securities, a custodian keeps the assets in accounts at central securities depositories, settles purchases and sales, collects income and processes corporate actions such as stock splits. For crypto assets, custody mainly means protecting the private keys that control the assets, often through cold storage, multi-signature arrangements and insurance, as described under crypto custody. Custody arrangements matter most when something goes wrong: properly segregated client assets should be returned to their owners if the custodian becomes insolvent, while assets mixed with the custodian's own may not be, as discussed under commingling.
"Custody" names the activity, a custodian is the firm that provides it, and "custody service" describes the commercial offering, which is why the full explanation sits under that name.