The agreement with which a person enters into civil society. The contract essentially binds people into a community that exists for mutual preservation. In entering into civil society, people sacrifice the physical freedom of being able to do whatever they please, but they gain the civil freedom of being able to think and act rationally and morally. Rousseau believes that only by entering into the social contract can we become fully human.
The problem of freedom is the motivating force behind The Social Contract. In the state of nature people have physical freedom, meaning that their actions are not restrained in any way, but they are little more than animals, slaves to their own instincts and impulses. In most contemporary societies, however, people lack even this physical freedom. They are bound to obey an absolutist king or government that is not accountable to them in any way. By proposing a social contract, Rousseau hopes to secure the civil freedom that should accompany life in society. This freedom is tempered by an agreement not to harm one's fellow citizens, but this restraint leads people to be moral and rational. In this sense, civil freedom is superior to physical freedom, since people are not even slaves to their impulses.
Strictly defined, a sovereign is the voice of the law and the absolute authority within a given state. In Rousseau's time, the sovereign was usually an absolute monarch. In The Social Contract, however, this word is given a new meaning. In a healthy republic, Rousseau defines the sovereign as all the citizens acting collectively. Together, they voice the general will and the laws of the state. The sovereign cannot be represented, divided, or broken up in any way: only all the people speaking collectively can be sovereign.
This is the executive power of a state, which takes care of particular matters and day-to-day business. There are as many different kinds of government as there are states, though they can be roughly divided into democracy (the rule of the many), aristocracy (the rule of the few), and monarchy (the rule of a single individual). The government represents the people: it is not sovereign, and it cannot speak for the general will. It has its own corporate will that is often at odds with the general will. For this reason, there is often friction between the government and the sovereign that can bring about the downfall of the state.
An abstract expression of the general will that is universally applicable. Laws deal only with the people collectively, and cannot deal with any particulars. They are essentially a record of what the people collectively desire. Laws exist to ensure that people remain loyal to the sovereign in all cases.
The will of the sovereign that aims at the common good. Each individual has his own particular will that expresses what is best for him. The general will expresses what is best for the state as a whole.
The sum total of each individual's particular will. In a healthy state, the will of all is the same thing as the general will, since each citizen wills the common good. However, in a state where people value their personal interests over the interests of the state, the will of all may differ significantly from the general will.
When Rousseau talks about the state of nature, he is talking about what human life would be like without the shaping influence of society. So much of what we are is what society makes us, so he suggests that before society existed, we must have been very different. In a different book, Discourse on Inequality, he speaks very highly of this prehistoric state, but in The Social Contract he is more ambivalent. In the state of nature, we are free to do whatever we want, but our desires and impulses are not tempered by reason. We have physical freedom but we lack morality and rationality. Still, Rousseau believed that this state of nature was better than the slavery of his contemporary society.
Civil society is the opposite of the state of nature: it is what we enter into when we agree to live in a community. With civil society comes civil freedom and the social contract. By agreeing to live together and look out for one another, we learn to be rational and moral, and to temper our brute instincts.
The common good is what is in the best interests of society as a whole. This is what the social contract is meant to achieve, and it is what the general will aims at.