The difference is that a nested class declaration that is also static can be instantiated outside of the enclosing class.
When you have a nested class declaration that is not static, also known as an inner class, Java won't let you instantiate it except via the enclosing class. The object created out of the inner class is linked to the object created from the outer class, so the inner class can reference the fields of the outer.
But if it's static, then the link does not exist, the outer fields cannot be accessed (except via an ordinary reference like any other object) and you can therefore instantiate the nested class by itself.