A delegate is a reference to a method that you can pass around as an object.
Imagine how useful it could be to have a method that allows its callers to provide part of its own logic. Every caller can have their own method, create a delegate (reference) to their method, and pass it to the method as a parameter. As long as the main method knows what arguments to pass in (if any), it can invoke the method by its reference (delegate).
Here's a simple example, specifically of usage #1 in your question:
void RemoveItem(string item, Action preRemoveLogic)
{
preRemoveLogic(); //we don't know what method this actually points to,
//but we can still call it.
//remove the item
}
void MyCustomLogic()
{
//do something cool
}
/* snip */
RemoveItem("the item", new Action(MyCustomLogic));
//I can pass a reference to a method! Neat!
Delegates are also very important for making events work in .NET.