Delegated administration lets you give limited admin work to people who are not full administrators, so a manager can handle users in a named role and the roles beneath it without owning the whole org. You define that slice in a delegated administration group: a set of users who share the same admin privileges, and not the same thing as a public group used for sharing. The group lists which roles they manage, which profiles, permission sets, permission set groups, and public groups they can assign, which custom objects they can customize, and whether they can log in as users they administer.