Plato's theory of Being and Becoming, and its relations to the forms, is rooted in the dichotomy between being and not-being. Prior to Socrates the Sophists, from Parminedes to Gorgias, had argued that because it was impossible by definition for Nothing to exist, it was impossible to describe or vocalize a negative state, and therefore also impossible to utter falsehood. "And now arises the greatest difficulty of all. If Not-being is inconceivable, how can Not-being be refuted? (Plato, Sophist) All that could be said must be somehow true, as false speech would not be speech and therefore could not be uttered. Being was arranged across the divide from an incomprehensible and/or impossible Not-Being. In addition, the nature of Being itself was somewhat suspect, as it was seen alternately as a great static or fluctuation One-ness, or as a multitude of ones; either position had flaws.
When Socrates/Plato arrived at a solution for the paradox of Being and Not-Being, their solution was that both Being and Not-Being were inherent in all things and defined not by negation of one another but by clarification of one another's properties, and that it then became apparent that the truth or reality of a thing is determined by the degree to which its professed or apparent nature lines up with what is otherwise the objective standard (be that metaphysical or experiential). This idea of a standard by which Being could be judged, apart from interaction with Not-Being, leads naturally to the idea that there are abstracted complex "forms" which are the sets of ideals or defining characteristics. The forms, by which the truth of other things can be judged, are abstract, complete, and relatively static. Actual reality, however, is concrete and incomplete, (the "falseness" of "not-being" fully ideal runs through all reality, corrupting it from the form) and therefore in a state of flux -- because of that, it is not actually Being at all, but Becoming. This clarification also is based on the Being/Not-Being debate, because that debate had suggested that Being was static and that falsehood and Not-Being were impossible partly because contradictory things could not both be true of one object (e.g. that it was both moving and at rest), so motion and change were largely illusory or were evidence of the fragmentation of reality. In response, the relationship between forms and functions came to be understood as a reaction between the overarching Being of archetypes and norms and the transitory Becoming of physical relativity. So to understand the significance of Being and Becoming in Plato's theory of forms, it is necessary to understand that the issue of Being and Becoming is in many ways the same as the issue of Being/Not-Being, and as such is bound up with issues of language and truth which make it fundamental to the theory of forms.
The preceding, rough explanation of the relationship between Being/Not-Being and Being/Becoming deserves a little further historical perspective before continuing with the examination of the topic. Plato did not develop this idea of forms in a vacuum -- in fact, in his drama/dialogue Parmenides, Plato shows how Socrates was given serious critiques and direction on his theory of forms by the older Sophist, whose intelligent questioning and rhetorical skill effectively dismantled the younger philosophers convictions for at least the space of that conversation. Socrates in all of the works consistently references other philosophers and schools of thought, the most notable of which would be the Sophists, (who were scribes and philosophers for hire) whom he vilified and also suggested were responsible for theories that there were no possibility for the existence of truths or falsehoods.
So Plato and Socrates had from these forerunners a heritage of thought which may have distorted their own vision to some degree. As the introduction to the Project Gutenberg edition of Plato's Sophist suggests, the idea that "no Being or reality can be ascribed to Not-being, and therefore not to falsehood, which is the image or expression of Not-being. Falsehood is wholly false; and to speak of true falsehood, as Theaetetus does (Theaet.), is a contradiction in terms...The fallacy to us is ridiculous and transparent... It is a confusion of falsehood and negation, from which Plato himself is not entirely free." Yet this was a vast, overarching preoccupation among philosophers at the time, and much of what might now be considered somewhat absurd in the argument was at the time a very serious question of the questionable possibility of full human communication, or as to whether humans could truly affect the world around them and see it change, or if it was essentially unchangeable.
"It was to Parmenides contrary to the logos for any real change to take place at all. In particular, it was impossible for a One Being to become many. For if the unity and being of the One are taken seriously, the One cannot in reality become other than what it already is -- no manifold world can actually proceed out of the One, no opposites actually exist to transform into each other. Therefore plurality, becoming, change, motion, flux, and so on, are not real, despite what our senses may lead us to believe." (McFarlane)
Plato would take this idea of the united unchanging One, and agree that it applied to Being. However, he synthesized this idea with an earlier theory by Herclitus which suggested that though the world is one, it is one in eternal flux. "All Flows," he suggests, and in this there is a perpetual cycle of life and death, of seasons, of moving and falling, and so forth. This constant flux was driven by the transformations and dualties of opposites, "the flux of existing things is characterized by the transformation between pairs of contrary principles. ... everything is One through the dynamic transforming of opposites into each other. Yet, Heraclitus sees structure in this flux." (McFarlane) Plato would suggest that both of these theories were true to some degree -- that while what he would call Being is indeed immune to change, that which he called Becoming was in constant flux. The world of forms (which is the manifestation of the One) is essentially unchanging, the expression of those forms in the world of shadows (which is the human reality) is changeable and mutable. To refer to the metaphor of the cave, one might add that the fire light flicker cast moving shadows on the wall when the form objects may not have moved.
This idea was backed up by a sophisticated and somewhat sloppy argument regarding transformation over time, and how at every individual moment in time the transforming thing is not actually transforming, it either is or is not in a given state. So things do not flux, when viewed abstractly frozen in time, but when they come to life through the intervention of time (or other beings) they begin to Become. So the Being self, which is an abstract form, is different from the experientially bases Becoming Self. That deformation from the One Being to the One Becoming is an idea that was unique to Socrates at the time, however it is not an idea that could have spring up without the intervening arguments regarding Being and Not-Being.
As was mentioned earlier, Parmenides and his sophist contemporaries suggested a number of variations on the theme that there cannot be a thing which is not. This is inherent in the "numbering" nature of language itself, which is to stay that in speech one always refers to the Not-Being thing as a thing, or an it, or some other object-term which implies selfhood. It is inappropriate and self-contradictory to say that there is a thing which is not a thing, or that there does exist "some thing which does not exist." This is because the negation of being assumes a being to be negated, and therefore (by simultaneously performed an assumption and destruction of being) is meaningless.
Plato proposed that the way around this dilemma was not to consider Not-Being as actually a form of negation, but rather as a form of relationship. That is to say that "Not-Being" is a particular variety of being, even as injustice is a form of justice, or poor grammar still a form of speech. Not-being is, precisely, the Other of Being -- it is everything which lies outside the form-ulaic definition of the Being thing, which is necessary to have be as well in order that the Being thing can be seen in relief. The Other, "Not-Being," thing can include very many or very few variations. The "Not-Being" of alive may only be dead or unborn, but the "Not-Being" of white can include the entire rainbow. Being is far more specific, one quickly notices, than Not-Being. However, it also cannot be understood without understanding Not-Being. White, for example, has no real meaning if there are not other colors to compare it to which are not-white. To quote Plato directly on this subject: