Leviathan takes place in a time of historical and philosophical change. Historically, it was written just before England plunged into civil war - the result of a bitter power struggle between the British Parliament and the monarchy. Hobbes' earlier work came down decidedly on the side of the royalists - a position that sent him into exile in France - but Leviathan was attacked by both sides. Those opposed to the monarchy saw an extension of his previous arguments, which in many ways it was, that the subjects of a commonwealth were to obey their ruler absolutely. Yet royalists were also offended by the work, despite the fact that Hobbes argued that a monarchy is a preferable form a government to a democracy, because they perceived attacks on a monarch's divine right to rule, as well as the rules that governed succession, and accused Hobbes of "switching sides."
Additionally, Hobbes was an avid student of the classics, and was convinced of the inherent instabilities of the ancient Greek and Roman democracies. When the struggle for power emerged in his own time, he undoubtedly saw this as another chapter in the story of democracy's volatility. For Hobbes, in the past democratic governments had only encouraged factionalism and in-fighting, which served only to distract it from more pressing issues, particularly invading armies.
Philosophically, Hobbes' era could be thought of as the beginning of the Enlightenment. He directly attacked the Aristotelian philosophy that was canonical in most universities, and was strongly influenced by the scientific achievements of Galileo and Kepler. Through them he came to believe in the centrality of motion in the universe, and sought to remake an entire philosophical system from the ground up based on these observations of motion.
Aside from their scientific achievements, Hobbes also admired Galileo and Kepler's faith in human reason, and their willingness to advocate near-heretical ideas (in Galileo's case, he was eventually excommunicated). This made Hobbes all the more willing to rethink the thoughts and concepts prevailing in his time, and he urged his readers to use their God-given rational capacities rather than defer to history and tradition. As his time in exile and numerous controversies showed, this was an idea held to in both principle and practice.
Perhaps most important are his contributions to Enlightenment philosophy. While thinkers of this time clamored for democracy rather than monarchy, Hobbes helped to introduce ideas of man's natural equality, representative government, and rationality. His social contract and description of human nature directly influenced Locke and Rousseau, who both used this method in their own theories of man and government. While they may have differed with Hobbes on some points, all three thinkers espoused the same belief that man's way of life in the state of nature is untenable, and hence a government must be erected to protect human life.
Hobbes current legacy remains disputed, just as his thought during his own time was as well. Almost any class on political theory will include a reading of Leviathan, yet in this modern age where it is almost taken for granted that democracy is the best and only legitimate form of government, one must wonder why he is still relevant. He is not just read in order to be refuted, but is read because so many of the main ideas of the Enlightenment that helped spawn the American and French revolutions can be traced back to him.