Skepticism is the old strawman and Nihilism is the new strawman. The strawman is the thing that everyone attacks but does not really take seriously. It used to be almost every philosophy book would start out by attacking skepticism. But then David Hume came along and Kant took him seriously and Philosophy got a lot more complicated as a result and took an Idealistic spin in order to solve the problems raised by Hume, and suddenly Skepticism was no longer the strawman because it was realized that the overcoming of Skepticism could actually boost philosophy into a new level, and so skeptics were taken seriously. Besides that Hegel based this modern dialectics on the work of Sextus Empiricus (a classical skeptic) as a way of understanding ancient dialectics. Nihilism appeared as the new scape goat, but of course is worse that skepticism. Gorgias becomes the new key figure (http://en.wikipedia.org/wiki/Gorgias) which is the closest thing to a nihilist in the ancient world. People like Gorgias who deny everything are called Academics in Skepticism. They are seen as the opposite of Dogmatics who affirm something about invisibles like Parmenides for instance. Because Nihilists appear in Skepticism it is possible to draw a distinction between them. For Plato, these various characters appear in Dialogues. In many of these Socrates becomes the main voice to which others merely agree. But in some poems other characters take the lead, like the Sophist, Parmenides, and the Laws. In these dialogues the Author and Narrator are hidden behind a veil, and we feel as if we are listening to a conversation between the characters, with Socrates being the hero of the piece. We are just so lucky to have these works whose meanings we are still plumbing. Of all these characters it is Gorgias who comes the closest to being a real nihilist. He like Nagarjuna negates everything. But Gorgias does so just to show off his powers of rhetoric, whereas Najarguna has the purpose of pointing at emptiness. Gorgias makes the radical claim:
The argument has largely been seen as an ironic refutation of Parmenides’ thesis on Being. Gorgias set out to prove that it is as easy to demonstrate that being is one, unchanging and timeless as it is to prove that being has no existence at all.”
“Gorgias begins his argument by presenting a logical contradiction, “if the nonexistent exists, it will both exist and not exist at the same time” (B3.67) (a violation of the principle of non-contradiction). He then denies that existence (to on) itself exists, for if it exists, it is either eternal or generated. If it is eternal, it has no beginning, and is therefore without limit. If it is without limit, it is “nowhere” (B3.69), and hence does not exist. And if existence is generated, it must come from something, and that something is existence, which is another contradiction. Likewise, nonexistence (to mê on) cannot produce anything (B3.71). The sophist then explains that existence can neither be “one” (hen) or “many” (polla), since if it were one, it would be divisible, and therefore not one. If it were many, it would be a “composite of separate entities” (B3.74) and no longer the thing known as existence.” http://www.iep.utm.edu/gorgias/
We know that Parmenides, as has been said in other answers, has three ways: Being, Appearance and Non-Being (existence). Parmenides denies all the other ways than Being. And Zeno develops arguments that show that if there is motion then there is argument. Gorgias used Paradox actively in his arguments, and appealed to Aristotle’s law of non-contradiction to prove his points. We have shown tentatively that these three ways line up with the Existentiels of Heidegger which are Rede (Chatter, Opinion, Appearance), Befindlichkeit (Existence, what is found), and Verstehen (intelligibility equals Being for Parmenides). Now when we look at the denial of Gorgias we see:
to me on (non-being) as a statement and this is the same as existence. (befindlichkeit)
The next denial is concerning knowledge which relates to verstehen and intellligibility which is the identification of thought with Being. He is denying intelligibly a basis characteristic of Being. Since knowledge is the most stable of anything in our experience, this is a big denial. It is also a denial of learning the process of attaining knowledge. (verstehen)
Next he denies communicability of knowledge, even if it existed. Communication as it was known in the Ancient world was talk (Rede).
Finally Gorgias denies the incentive to communicate. This seems to go beyond the three Existentiels. But we must remember that the mysterious Goddess that is teaching Parmenides is probably Persuasion. Persuasion is based on Rhetoric, and it is precisely the attempt to plant an incentive in the hearer.
Heidegger says that the core of the Existentiels is Care. To have care is to have incentive, to have within oneself rather than to be persuaded. Thus Dasein is authentic to the extent that he extracts himself from the They (mitsein) to concentrate on its own cares in the face of death.
So where Gorgias exceeds the ways of Parmenides gives us more information about the Existentiels that we would have otherwise, and makes us remember the Door which at which Parmenides meets the Goddess, whom some have said is probably Persuasion. In Persuasion you care what others do and you want to give them incentives to act the way you think they should act. Basically you care enough to make an argument attempting to persuade them.
Now lets look at these four ways, and see if they are not like the four ways that Plato discusses in his Timaeus and the Sophist.
In each beginning of the Timaeus there is a different kind of Being that is indicated. In the beginning the contrast is between Parmenidian stasis and Heraclitian flux. But when Plato begins again he talks about a third kind of Being which is synonymous with Hyper Being or Difference of Derrida, or -B-e-i-n-g- crossed out of Heidegger. When Plato begins again he does not talk about a fourth kind of Being but he is talking about nature and if read carefully we can see in it hints of the fourth kind of Being in the text when we contrast it to the other beginnings.
Notice that we might compose an interleaving as follows:
beings, men of earth
befindlichkeit, existence, to me on, Defense of Palamedes, means did not exist to do permit treason, System of things.
Pure Being, Parmenidian Stasis, greater mystery
verstehen, intelligibility, thought, knowledge as process of learning (dissipative ordering SS)
Process Being, Heraclitian Flux, lesser mysteries
rede, communication, chatter, nihilism, inauthenticity (autopoietic SS, cf Luhmann)
Hyper Being, Differance, -B-e-i-n-g- crossed out
incentive, persuasion, inclination, tendency, Care dispoistion (reflexive social SS)
Wild Being, Hierophant who initiates
Goddess at Gate, door to meta-system (DUAT). Helen child of Zeus and Leda (defended, we are persuaded of her innocence in the Encomium)
Ultra Being, God of the cult being initiated into.
So what we are seeing here when we compare Gorgias, Parmenides and the Existentiels of Heidegger is the correspondance with the Special Systems.
This is quite unexpected.
Gorgias brings the Antimony to Parmenides and in that preserves the structure of his own argument, and allows us to undersand more than we would have otherwise. His Academic nihilism is merely the inversion and negation of the the positive argument of Parmenides the Dogmatist. To this Sextus Empiricus introduces a third way which is similar to Buddhism in many ways. Basically the skeptic wants to keep the dialectic going at all costs and will even say things he does not really believe to keep the argument going. The skeptic is content in the process of the argument of the dialectic continuing. The Skeptic is the one who like Kant is critical of the Dogmatist and the Academic both. But since he will say what he does not really believe to keep the argument going he is seen as a threat to serious philosophical argumentation.
In general the Skeptic is the one whose words cannot be believed, and so you never know what he believes because he will keep the dialectic going at all costs, and he finds peace in that exploitative process, as long as it continues. He does not believe it will reach a conclusion, but is satisfied that the participants are trying.
The Nihilist on the other hand is one who is an academic and so will deny anything like Gorgias just for the heck of it. He is the one who has lost all meaning in his world, ultimately. Nihilism ends up with Alienation, Anomie or Madness. For instance, the obsession with Nihilism in Will to Power eventually drove Nietzsche insane. Achilles on the other hand loses meaning when he sees that the Acheans are no better than the Trojans (both steal women), and his response is doing nothing in the midst of war (alienation from his comrades), and then Berzerker Mode once his friend Patroclus is killed (Madness). Alienation sometimes results in serial killings, suicides that take others with them, because the blame is projected outside the individual and exists in society.
Deleuze and Guattari deal with socially induced madness in Capitolism and Schizophrenia trying to understand it and create a philosophy at the level of Wild Being.
Badiou (http://en.wikipedia.org/wiki/Badiou) [Being and Event, transl. by Oliver Feltham; (New York: Continuum, 2005)] and Zizek (http://en.wikipedia.org/wiki/Zizek) in their own ways are going beyond this to try to understand Ultra Being. In Badiou the ultra-one is the ONE which appears out of the Multiple, and produces the countable ones. The Multiple is the absoute heterogeneity that Deleuze did not attain for all his talk of difference [Deleuze: The Clamor of Being, transl. by Louise Burchill; (Minnesota University Press, 1999)]. Zizek on the other hand uses Lacans (http://www.iep.utm.edu/lacweb/; http://en.wikipedia.org/wiki/Jacques_Lacan) idea of jousissance and other tropes to describe singularities within the field psyche expressed by the registers Imaginary Symbolic and Real that form a Borromean Ring (http://en.wikipedia.org/wiki/Borromean_rings).