Rousseau's Social Contract was an attempt to explain how societies can function. He suggested that people implicitly agree to a contract that restrict their actions (eg. not to kill each other, respect each others' property, etc.), rather than requiring a King to keep order (like Thomas Hobbs).
Yes. By giving up the right to kill you gain the right not to be murdered. And so on down the line. This is why it is a contract between people rather than an enforced state such as the hobbsian or the plationic model. A later development, the utilitarian state, never really took off, though it was preferable: you should so order your affairs so as to incur the greatest good for the greatest number. Sadly Adam Smith seems to run the world these days.