The Axiom of Empty Set asserts that a set exists which is empty. It does not assert the uniqueness of this set; using the other axioms of Zermelo-Fraenkel set theory, this set can be shown to be unique.
This axiom is sometimes called the zeroth axiom of set theory (see Zeroth Law of Thermodynamics) because it states something so obvious that it's easy to forget to formally state it as a rule.
The Axiom of Empty Set and the Axiom of Choice are the only two axioms of set theory that non-constructively assert the existence of a set.