Delegates, to my understanding, provides a way of specializing the behavior of a class without subclassing it.
Some classes have complex generic behavior, but are still meant to be specialized. Think of a Window class in a GUI framework: A Window can propably do a lot on it's own, but you would most likely still want to specialize it in some way. In some frameworks, this is done via inheritance. A different way of doing it is with delegates. Say you want something to happen when the Window resizes: Your delegate class can then implement a method called onWindowResize (provided of course that the Window class supports this), which gets called whenever the Window resizes and is responsible for any specialized behavior when the Window resizes.
I'm not going to argue the merits of delegation over inheritance, but suffice it to say that there are many who feel that delegation is "cleaner" than inheritance.