Delegation in Project Server and Project Online is an extremely powerful capability that can be used in numerous situations. For support situations, help desk personnal can use it to login as someone experiencing a permissions-related issue to confirm, research and resolve an issue. In a project team situation delegation can be used to submit time for a Team Member who is out sick or on vacation, or to accept time for a Project Manager who is away attending training.
This training video is key to gaining a solid understanding in how to leverage the Manage Delegates feature in Project Server & Online. In addition to Managing Delegates, once created a user next needs to Act as Delegate in order to work on behalf as another userr. Therefore be sure to watch session 2 of 2 on Delegates: Act as Delegate, to complete your understanding of this very useful functionality.