The first category, Non-Being, is essentially an empty set as nothing does not exist. The only “things” which could be properly said of this category are things which could not possibly exist in any universe, e.g., contradictions. If we try to conjunct the idea of a square with the idea of a circle we get a contradiction as the properties of the square cancel out, and are canceled out by, the properties of a circle. Thus, a square circle, due to its contrary properties, could not possibly exist in any possible universe. This means that square circles are not things at all; they have no possible being. Thus, any “thing” of this sort is no thing at all and would therefore being an example of what we have in mind when we speak of the realm of Non-Being.
Becoming is that category or set of things that exist, but in some limited way. Anything which exists but is limited by time, space, shape, color, or any other property which is necessary or accidental to that thing, would be properly categorized as part of Becoming. Thus, this set of things obviously includes everything in the material universe as everything made of matter is characterized by its place in the universe (i.e., space), its duration of existence (i.e., time), its mass, its motion, its position relative to other things, etc.
Being is the set which includes existence which is not limited in any way. This is the set of that which is without qualification; it exists, full-stop. Being cannot be said to exist anywhere or anywhen or having any qualities that would limit or distinguish it from any other existence.
In the History of Philosophy we can see these ontological distinctions in the work of Herakleitos, Parmenades, and Plato (among many others).
Herakleitos held that the observable world was primarily characterized by becoming as it is in a constant state of flux. Parmenades, on the other hand, held that change (whether temporal or spacial) was an illusion and that trying to conceive of change ultimately leads to unresolvable contradictions.
Plato attempts to synthesize the views of the pre-socratic philosophers by claiming that both Being and Becoming are real aspects of the universe, but form distinct dimensions of reality (thus we think of Plato as a metaphysical dualist).