Delegation allows you to execute methods (and optionally pass parameters) "backward." For example, if you have one class that references another class (parent and child), the child can fire back to the parent without referencing the parent.
Notification Center, on the other hand, listens and waits until it hears the message it is waiting for. This allows for multiple listeners in multiple view controllers to wait and listen for a given message.
As for relations, delegation is a 1 to 1 relationship while Notification Center is a 1 to 1/many relationship.