On which object can an administrator create a roll-up summary field?

A roll-up summary field can only be created on objects that are on the master side of a master-detail relationship. A master-detail relationship is a special type of relationship between two objects where the detail (child) record's lifecycle is tightly bound to the master (parent) record. This means that the detail record cannot exist without the master record, which sets up a scenario where summarizing data makes sense.

In a roll-up summary field configuration, the administrator can define calculations such as COUNT, SUM, MIN, and MAX based on the related detail records. For example, if you have an Account (master) and related Opportunity records (detail), you can create a roll-up summary field on the Account object to display the total amount of all its Opportunities.

The other options refer to types of relationships where roll-up summary fields cannot be created. In a lookup relationship (which is less tightly coupled than a master-detail relationship), there is no ownership of the detail records by the parent, preventing the summarization of data. Hence, options referring to parent or child sides of lookup relationships or the detail side of master-detail relationships do not apply for roll-up summary fields.