What is mailbox delegation?
Similar to having an assistant that helps you manage your incoming paper mail, another person, known as a delegate, can receive and respond to email messages and meeting requests and responses on your behalf. You can also grant the delegate additional permissions to read, create, or change items in your Microsoft Exchange Server mailbox.
What does Delegate Access do?
Delegate Access goes beyond just sharing access to your folders. Delegates are granted additional permissions, such as creating email messages or responding to meeting requests on your behalf.
Additional details here: Microsoft - What does Delegate Access do?
What permission levels are there?
With this permission, the delegate can read items in your folders.
With this permission, the delegate can read and create items, and change and delete items that he or she creates. For example, a delegate can create task requests and meeting requests directly in your Task or Calendar folder and then send the item on your behalf.
With this permission, the delegate can do everything that an Author has permission to do and additionally can change and delete the items that you created.
See Related Links & References for additional supporting in assigning a delegate or setting up a delegated account.