In Chapter 1, Locke first reiterates his arguments from the First Treatise against Sir Robert Filmer's writings. His points refute Filmer as follows: Adam was NOT given absolute authority over the world and his children by God Adam's heirs, therefore, did not have this authority No one can claim rights since it is impossible to identify Adam's heirs today.
Locke aimed to refute Filmer's theory of the divine right of sovereignty. Locke finishes the chapter by noting that one must not confuse different types of power--paternal, familial, and political--for each has very different characteristics. He defines political power as the right to make laws for the protection and regulation of property; these laws are backed by the community, for the public good.
Locke addresses the natural instincts of people, or the state of nature, in order to define political power. In Chapter 2, Locke explains the state of nature as a state of equality in which no one has power over another, and all are free to do as they please. He notes, however, that this liberty does not equal license to abuse others, and that natural law exists even in the state of nature. Each individual in the state of nature has the power to execute natural laws, which are universal. Locke then posits that proof of this natural law lies in the fact that, even though a person cannot reasonably be under the power of a foreign king, if a person commits a crime in a foreign country they can still be punished. Locke states that natural law simply demands that punishment fit the crime--a person in the state of nature can redress any crime to discourage the offender from repeating it. Locke concludes by noting that all people are in a state of nature until a special compact or agreement between them (which he promises to describe later) makes them members of a political society.
Commentary
In the Second Treatise, Locke rises above the specifics of the political situation described in the Introduction to outline a coherent theory of liberal political government, based on the sanctity of individual property and the state of nature. In Locke's state of nature, no person has control over another, natural law governs and renders all people equal, and every individual holds the executive power of natural law.
Locke's theory includes many assumptions. First is the assumption of a system of morality--the natural law derives from a theory of justice, a set of rights. No one would have any "rights" at all in the absence of a moral code applicable to human actions, nor would there be any standard of "just" punishment. Locke frequently uses the term "rights" and appeals to conscience and "calm reason", all of which reflect his assumptions about justice and morality.

Since this is our first encounter with the text, let's take this as an entry-point to talking a bit about Locke's writing itself, his syntax, word choice, and so on. Oddly enough, John Locke, the great Treatise writer and political philosopher, had a fairly cautious relationship with language. Book III of his very famous Essay Concerning Human Understanding is all about language, and expresses the idea that language should only be used to convey ideas and meanings as simply, clearly, and economically as possible. Locke's Second Treatise is actually quite simple to read, because he moves directly from point to point, and is not given to hyperbole or repetition. Locke is not a rhetorician; rather than play with words or language to compel the reader, he states his ideas as strongly and clearly as possible. He notes in the Preface that "railing" responses to his work will not refute his ideas--only rational arguments will. This simplicity of Locke's writing gives his ideas a sense of compelling empirical clarity, which can often cause one to overlook flaws in his philosophy (which we will address later).