Delegated task is a task, not related to any process, created by one user and delegated to other user or users.
Any user can to create Delegated tasks.
If you create a Delegated task for other users you will have full access to it:
View the task in "Delegated tasks" list on Tasks screen
Edit and delete the task
Change task due date
Set or change task assignee
Complete and Reopen the task
If you are the candidate assignee for Delegated task you can:
View Delegated task in "Available tasks" list on Task screen
Get this task to work on it (assign to yourself)
If you are assignee of Delegated task you can:
View Delegated task in "My tasks" list on Task screen
Change task due date
Unassign it from herself
Complete and Reopen the task
If you mentioned in Delegated task comments and is not participated (you are not task creator or assignee) you can view Delegated task by click on task link in email notification.
All other users have no access to Delegated tasks.