What are the advantages of delegates in C#.NET?

Delegates provide additional level of abstraction, because you can abstract out some action expressed in the code and pass is as a parameter in some other method, like any other parameter.
This is called "first-class citizen".

OOP virtual methods and late binding does something like that (and this is also a method parameter, hidden on), but delegated has no limitations: delegate instances can accept any method in its invocation list; if the method has matching signature it can be virtual or not, static or instance method, of the same class or any other (which is the main difference between a delegate and a virtual method).

To show the main purpose of first-class citizen conception, imaging you have a complex data traversal algorithm. You want to select only a sub-set part of data members by some predicate and perform some action on each. The traversal algorithm should be universal, agnostic to both action and predicate. Here is how:

void TraverseData(
         MyDataContainer data,
         Func<DataElement> predicate,
         Action<DataElement> dataProcessor) {
    //...
    DateElement element = //...
    if (predicate(element))
        dataProcessor(element);
    //...
} //TraverseData

[EDIT — fixed reference]
See also my other answer: What is the Extra Advantage of Delegate[^]

[EDIT]
For advanced use of delegates, see my new article on the topic:
Dynamic Method Dispatcher[^]

—SA