Nature
The Discourse on Inequality clarifies that it cannot answer the question of what causes inequality without answering the question of the role of nature itself. Nature appears in two ways in the Rousseau's essay. First, it is a powerful image of what man's life was like before the corrupting effects of society. Rousseau believes that though man's life was difficult, he was nonetheless free, and that he was therefore happy. At the same time, nature, as Rousseau clarifies in the preface, is the ultimate criterion of legitimacy for social institutions. Since there were no governments in nature, no governments are truly legitimate, since they were all derived from inequality. What man did have in the state of nature was feeling—he pitied others, and he feared for himself. Rousseau considers these feelings to be the only legitimate basis for man's social life. Therefore, in order to be better citizens, we have to cultivate our feelings, rather than argue about which form of government "works" best.
Freedom
The question of what it is to be free is a persistent one in the Discourse on Inequality. Rousseau believes that man cannot be happy without being free. Indeed, the only way for man to be happy is to be free. But what is freedom? In the American constitution, for example, the government grants individuals certain freedoms—freedom of speech, freedom of religion. Essentially, the government promises not to intervene in or to infringe on rights that the individual possesses. Rousseau disagrees with this view. For Rousseau, freedom consists only of independence—in so far as the individual is dependent on others (the government, say) for his freedoms, they aren't really freedoms. The only freedom is in not being told what to do, and in not telling others what to do.
The Corruption of Society
The Enlightenment, the intellectual period in the mid-to-late 18th century in France, England, and to a lesser extent Germany, believed that human beings could improve themselves and the world by embracing culture and science, and rejecting the superstitions of religion. The period was an unprecedented flowering of both, and of democratic uprisings against monarchies, of which the American and French Revolutions are two important examples.
But Rousseau was part of what has sometimes been called the Counter-Enlightenment: he rejected, or was at least deeply skeptical, of the notion that any social institution could improve man. All society necessarily alienated man from his true feelings, and forced him into a state of dependence on others, whether for sustenance or for esteem. And that made him unhappy. Society was effective at creating solutions only for the problems it had created in the first place. Culture and science, however impressive they might be, only made man softer, and substituted false feelings and useless knowledge for the only that truly mattered, insight into himself. This view baffled Rousseau's readers, who wondered why such a person would bother writing philosophy in the first place. But that sense of intractable contradiction, the pain of living in a world one doesn't accept as legitimate, is central to all of Rousseau's thinking—he thought that the primacy of that feeling shouldn't be sacrificed so that a "solution" could be found for it. This theme is echoed in the centrality of the concept of "self-perfectibility"—man is able to improve himself, but at the cost of his own happiness.
Authority
Another central question of the Second Discourse is how human beings could have accepted any authority over themselves if they were happy as free individuals. In Rousseau's time, there were many different explanations for where authority (the legitimacy by which the king could command his subjects, or a parliament could pass laws) came from—God, reason, practical necessity, or the natural superiority of some human beings over others. Rousseau's reflections on the state of nature, in which all people were equal because no one had need of any one else, suggest that all forms of authority are, in a sense, illegitimate, because they are rooted in the inequality between the governed and those governing them. No form of government based on this inequality can claim to be truly legitimate. This would be one of the animating concepts of the American and the French Revolutions.
Inequality
As the discourse's title suggests, Rousseau is concerned above all with the question of inequality. Where does it come from, and can it ever truly be regarded as legitimate? The going view in Rousseau's time was yes. Simply, people were unequal, they'd always been unequal, and some force, whether God or nature, had made them that way and kept them that way. Rousseau rejects this view, and the imagination of a state of nature is meant to serve as a powerful rhetorical tool to the reader to bring home just how unnatural the inequality of Rousseau's time was, and just how much anguish it would bring to a "truly free" person—i.e., not the reader. For Rousseau, human history is the history of the compounding of inequality, and the creation of tools and structures to present it as legitimate. In this sense, human history has been one disaster and mistake after another. Over the generations, these mistakes have been codified in our institutions, and worse, in our minds, gradually getting us to accept what Rousseau perceives as a slavery that is completely foreign to human nature.
The Social Contract
If the original human beings were free, why would they ever have wanted to enter into something as degrading as society? This is the chicken-and-egg question that Rousseau faces. Rousseau rejects two going answers to this question. The first is that the strong conquered the weak. Rousseau rejects the idea that any truly free person would ever consciously give away their freedom. The conquered would have killed themselves first, Rousseau believes. Second, that the weak banded together to protect themselves against the strong. Here, too, Rousseau believes that the slavery of dependence on others would have dramatically outweighed whatever personal protections society could afford for the wealthy. He believes that the rich and the poor, that is, the strong and the weak, must have entered into society together, because the inequality that existed between them caused perpetual strife. Thus the first formalized social arrangements were then made in the mistaken belief that they were preserving freedom for both parties, when in fact they were destroying it.
Animals
Like many philosophers before and after him, Rousseau is fascinated by animal life—by the many similarities it shares with human life, and with the task of trying to identify the crucial difference. Rousseau is strongly tempted by the idea, which echoes the writing of the French Jansenist philosopher Blaise Pascal, that what makes man superior to animals also makes him unhappy. Man is both the highest being in creation, and for that reason, also the lowest, because his mind makes him strive for things he can never attain, and sets him to work on all sorts of useless projects. Because he has a mind, he is aware of the limitations of his own life, which animals are not. Animals serve as a model of what human life would be like without the capacity for self-perfection.
