Equality means identicality with regard to natural or divine rights. For instance, a person who has only the natural or divine rights to life, liberty, and the pursuit of happiness would be equal to another person who also only has those rights.
Natural right is the obverse of positive right. “Positive right” means simply “posited right”, in the sense that the right in question is posited (created) and not natural (discovered). Divine right means a right posited by God. However, “positive right” typically means “right posited by human beings”. As God is the creator of the whole of nature, any rights he posits are by necessity natural rights.
Natural right depends on natural ends. To say that man has the natural right of liberty is to say that man’s liberty is an essential part of his nature, of his natural state. Thus in the Book of Genesis, God is recorded as saying “it is not right for man to be alone”. This means man’s being together is an essential part of his nature, of his natural state (in the Book of Genesis: his Paradisal state). Note that God first created him alone, yet this beginning is not his natural state; man only becomes natural when he reaches his natural state, his natural end (“end” in the sense of “goal”). But this means that even and especially when he has not reached it yet, he has the natural right to it—in this example, the natural right to be together.
Now the reason Nietzsche is against equality is that he disagrees with egalitarians’ conceptions of man’s natural end. Thus in the example of liberty, he completely disagrees with egalitarian conceptions of freedom:
“[W]hat is freedom! That one has the will to assume responsibility for oneself. That one maintains the distance which separates us. That one becomes more indifferent to difficulties, hardships, privation, even to life itself. That one is prepared to sacrifice human beings for one’s cause, not excluding oneself. Freedom means that the manly instincts which delight in war and victory dominate over other instincts, for example, over those of ‘happiness’. The human being who has become free—and how much more the spirit who has become free—spits on the contemptible type of well-being dreamed of by shopkeepers, Christians, cows, females, Englishmen, and other democrats. The free human being is a warrior.—” (Source: Nietzsche, Twilight of the Idols, “Forays of an Untimely Man”, section 38.)
This is basically the same difference as between the Superman and the Last Man. Man’s natural end as dreamed of by egalitarians is the Last Man; man’s natural end as seen by Nietzsche is the Superman. But not everyman is capable of becoming a Superman. Rather to the contrary, only the fewest are capable of that. The right road for the rest does not consist in becoming Supermen themselves, but for example in becoming—biological or spiritual—fathers or forefathers to Supermen, or even in being “sacrificed” for the “cause” that is the Superman. And actually, not everyman is capable of becoming a Last Man, either! Thus Nietzsche’s Zarathustra says about the Last Men:
“No shepherd, and one herd! Everyone wanteth the same; everyone is equal: he who hath other sentiments goeth voluntarily into the madhouse [or, as Leo Strauss put it, to the psychiatrist].” (Source: Nietzsche, Thus Spake Zarathustra, “Zarathustra’s Prologue”, section 5.)
Not everyman is capable of having his head shrunk enough to fit in with the herd. And indeed, Nietzsche says:
“The criminal type is the type of the strong human being under unfavorable circumstances: a strong human being made sick. He lacks the wilderness, a somehow freer and more dangerous environment and form of existence, where everything that is weapons and armor in the instinct of the strong human being has its rightful place. His virtues are ostracized by society; the most vivid drives with which he is endowed soon grow together with the depressing affects—with suspicion, fear, and dishonor. Yet this is almost the recipe for physiological degeneration. Whoever must do secretly, with long suspense, caution, and cunning, what he can do best and would like most to do, becomes anemic; and because he always harvests only danger, persecution, and calamity from his instincts, his attitude to these instincts is reversed too, and he comes to experience them fatalistically. It is society, our tame, mediocre, emasculated society, in which a natural human being, who comes from the mountains or from the adventures of the sea, necessarily degenerates into a criminal. Or almost necessarily; for there are cases in which such a man proves stronger than society: the Corsican, Napoleon, is the most famous case.” (Source: Twilight of the Idols, ibid., section 45. Cf. section 13 of the first treatise of On the Genealogy of Morals: “birds of prey” are not free to become “lambs”.)
Now it’s also Napoleon whom Nietzsche uses as a “metaphorical” example of the Superman in section 48 (cf. section 57 of The Antichrist). Only the strongest have the “right”, i.e. the might, to enter man’s natural state:
“The customary understanding suggests that to enter the esoteric all that is needed is permission or instruction and one can walk on in. Nietzsche’s correction suggests that the esoteric view is unattainable or inaccessible to anyone who is not the kind for it: no one can be carried to the view from the height.” (Source: Laurence Lampert, Nietzsche’s Task, pp. 72-73.)
Both for Nietzsche and for modern egalitarians, everyman has permission to enter man’s natural state; in other words, everyman has the opportunity. But not everyman has the means. The egalitarian dream is that everyman can gain “Paradise” if he only puts himself to it. Nietzsche shatters that dream by pointing out the evident fact that people differ drastically with regard to potential and thereby to value—not just economically (Adam Smith), but in the overall competition that is life.
“In old Russia, the Czar, each year, granted—out of the shrewdness of his own soul or one of his advisors’—a week’s freedom for one convict in each of his prisons. The choice was left to the prisoners themselves and it was determined in several ways. Sometimes by vote, sometimes by lot, often by force. It was apparent that the chosen must be a man of magic, virility, experience, perhaps narrative skill, a man of possibility, in short, a hero. Impossible situation at the moment of freedom, impossible selection, defining our world in its percussions.” (Source: Jim Morrison, The Lords.)