A delegated administrator is a normal SharePoint user that has been given the additional ability to manage users for a particular site.  Primarily, the key role of a delegated administrator is to invite other users to his or her site. 

To promote a user to a delegated administrator, from the site's Site Settings page click Manage Users in Authorization Store under the Delegation Management menu.

The management interface of the Manage Users in Authorization Store page allows you to add, update and delete users.

From the menu toolbar, click New and then Add New User.

Next, on the Add User to Authorization Store page, complete the new user's details include User Name, E-mail Address, Password, and a password recovery question.   When finished, click the Finish button to create the new user.

Next, return to the Site Settings page and click the Manage Delegate Administrators link under the Delegation Management menu. 

From the Manage Delegate Administrators page, click New from the menu toolbar and then select Add Delegate Administrator.

Finally, on the Add Delegate Administrator page enter the name of the user that was just created and then give the user a Delegated Role and permission set.  Click the Finish button to complete the process and promote the user to a delegated administrator.