"Third Party Application" AKA "Client" or "Relying Party"
This is the app attempting to obtain access to the details of the User's Account. Permission is required from the user in order to obtain access.
"The API" AKA "Resource Server" or "Identity Provider"
This is server(s) hosting the resources that are protected i.e. the system having access to the User's data.
"The User" AKA "Resource Owner"
This is the person authorising a third party application to access their account details.