Much of the power comes from the fact that an object can be referenced by a variable of the interface type. This is subtly vary powerful.
private foo()
{
IDemo demoOne = new MyClass1();
IDemo demoTwo = new MyClass2();
}
This can become vary powerful because you can encapsulate different behaviors. For example:
private foo(bool option)
{
IDemo demo = option ? new MyClass1() : new MyClass2();
}
private bar (IDemo demo)
{
demo.Show();
}
Now bar can use the IDemo object without having to know which concrete implementation of IDemo is passed in. The decision about which implementation to use is encapsulated in the foo method. This might not seem like a big deal in such a simple example. If you look at the links posted in tigger's answer, you will see where this can become very useful.
One case where this is particularly useful is with unit testing. You can have a business logic class that takes an interface to a data layer object. When the application runs, the business logic class is passed an instance of the real data layer object. When the class is unit tested, it is passed an instance of a object that returns test data. This allows the unit test to run with predictable data inputs. This is known at Dependency Injection.
Another useful case is when you want to interact with framework or third-party code. Let's say you want to implement a custom collection. If your class implements the IEnumerable interface, you can iterate through the items in the collection in a foreach loop. The framework doesn't need to know how your class stores the items or what is in the items, but if it knows that you implemented IEnumerable, it can allow you to use a foreach loop.