In Leviathan, Hobbes set out his doctrine of modern natural right as the foundation of societies and legitimate governments. In the natural condition of mankind, while some men may be stronger or more intelligent than others, none are so strong and smart as to be beyond a fear of violent death. When threatened with death, man in his natural state cannot help but defend himself in any way possible. Self-defense against violent death is Hobbes' highest human necessity, and rights are borne of necessity. In the state of nature, then, each of us has a right to everything in the world. Due to the scarcity of things in the world, there is a constant, and rights-based, "war of all against all" (bellum omnium contra omnes). Life in the state of nature is "solitary, poore, nasty, brutish, and short" (XIII).
But war isn't in man's best interest. He has a self-interested and materialistic desire to end war – "the passions that incline men to peace are fear of death, desire of such things as are necessary to commodious living, and a hope by their industry to obtain them" (XIII, 14). He forms peaceful societies by entering into a social contract. According to Hobbes, society is a population beneath an authority, to whom all individuals in that society surrender just enough of their natural right for the authority to be able to ensure internal peace and a common defense. This sovereign, whether monarch, aristocracy or democracy (though Hobbes prefers monarchy), should be a Leviathan, an absolute authority. Law, for Hobbes, is the enforcement of contracts. The political theory of Leviathan varies little from that set out in two earlier works, The Elements of Law and De Cive (On The Citizen). (A minor aside: Hobbes almost never uses the phrase "state of nature" in his works.)
Hobbes' leviathan state is infinitely authoritative in matters pertaining to aggression, one man waging war on another, or any matters pertaining to the cohesiveness of the state. It can say nothing about what any man does otherwise - so long as one man does no harm to any other, the sovereign should keep its hands off him (however, since there is no power above the sovereign, there is nothing to prevent the sovereign breaking this rule). In a word, Hobbes' political doctrine is "do no harm." His negative version of the Golden Rule, in chapter XV, 35, reads: "Do not that to another, which thou wouldst not have done to thyself." (Compare with the Christian golden rule, which encourages actively doing unto others: for Hobbes, that is a recipe for social chaos.) Hobbes is the founder of political liberalism (liberalism understood in a specific way.
Leviathan was written during the English Civil war; much of the book is occupied with demonstrating the necessity of a strong central authority to avoid the evil of discord and civil war. In particular, the doctrine of separation of powers is rejected: the sovereign must control civil, military, judicial and ecclesiastical powers.
In Leviathan, Hobbes explicitly states that the sovereign has authority to assert power over matters of faith and doctrine, and that not to do so is a recipe for discord. Hobbes presents his own religious theory, but states that he would defer to the will of the sovereign (when that was re-established: again, Leviathan was written during the Civil War) as to whether his theory was acceptable. Tuck argues that it further marks Hobbes as a supporter of the religious policy of the post-Civil War English republic, Independency.
The word "Hobbesian" is sometimes used in modern English to refer to a situation in which there is unrestrained, selfish, and uncivilised competition. This usage, now well-established, is unfortunate for two reasons: first, the Leviathan describes such a situation, but only in order to criticise it; second, Hobbes himself was timid and bookish in person.