Introduction to Philosophy/What is Political Philosophy
A good definition for Political Philosophy is found only after determining what is politics, which is a sticky question to begin with. Politics could be defined as "the question of how to distribute a scarce amount of resources 'justly.'" Which is, essentially, the way in which people obtain, keep, and exercise power. Political philosophy, then, is the study of the theories behind politics. These theories may be used to gain power or to justify its existence.
Mostly, however, they have been used to justify or legitimate the existence of contemporary political structures by appealing to "rationality," "reason," or, among others, "natural law."
Plato's Republic is a good starting point for political philosophy, however, it's really a treatise on education. It starts out by trying to define Justice (one of Kenneth Burke's "God Terms"). In it, he makes an argument for a sort of acetic life-style by, through a standard Platonic dialogue, laying out a minimally functional society. He then, somewhat parodically, responds to the question of luxury by outlining how to 'justly' lay out a state that will accommodate luxuries for the entitled (a state that looks very similar to Sparta). It's a good starting place, because it lays out his conception of Justice, which, inevitably, is based on his theory of the forms, which is a similar basis of conceptions of natural law.
Skipping a few thousand years and many important texts, we get to Nicolo Machiavelli's The Prince, which was written in 1513 and published, after his death, in 1532. Machiavelli lived in Florence under the Medici family's rule. During a brief period of reform, the Medicis were chased from power, and Machiavelli became a diplomat. When the Medicis returned, Machiavelli was basically exiled. One reason he might have written The Prince was to try to return to public life in Florence. This book is often criticized for its moral relativism, and, in an inadequate summarization, that power defines moral action.
Moving along, we can get to the social contract theorists, namely Jean Jacques Rousseau, Thomas Hobbes, John Locke, Charles Montesquieu, and Baruch Spinoza. This, of course, will be a brief and incomplete treatment of them, but it's a starting place. Hobbes' theory is mostly found in his book Leviathan. In it, he defines the state of nature (the prepolitical society) as a place where life is "nasty, brutish, and short." It's important to understand that Hobbes was writing after the Thirty Years War (a religious conflict between, primarily, the English Protestants and the Spanish Catholics), so he had a very pessimistic view of Human Nature. He basically thought that man, left to its own devices, would war against itself, hence the above quotation.
From that, Hobbes had a view that society with a state, at its worst, was better than having no state at all, so he concluded that any state action would be justified if for no other reason than that it is a modus vivendi, a lesser evil. The crux of this, and all social contract theory, is that the citizen has a sort of contract with the state in which people give up some autonomy to make their lives better. For Hobbes, this autonomy was given up to protect life at its most fundamental level.
Locke, however, has an entirely different notion. His view of the state of nature (the prepolitical society) is much nicer. Basically people will respect each other and not infringe on another's person or property. If someone does, then the agressee has the natural right to rectify the situation, and anyone else who witnesses an aggressor has the duty to help the agressee. Locke concedes that property disputes will eventually be numerous enough that this would be time consuming, and in the general interest, people will form a state in order to have someone else protect their property and persons--basically to settle disputes--out of convenience. Much of this is laid out in The Second Treatise on Government. It's important to note that much of the U.S. constitution is based on Locke's political philosophy.
A question, of course, is why these social contract theorists are doing what they're doing. I mean, they're creating these ridiculous constructs of a prepolitical reality through a strange process of abstraction. One answer is that they're justifying the existence of the state and the state's actions. Under Hobbes' view, the state is thus legitimate in anything it does, in any violation of 'human rights' because, well, things would be worse off without it. Locke goes about it in order to have a sort of neutral procedure to protect property rights.
More contemporary social contract theorists include Robert Nozick (his main book on this, Anarchy, the State, and Utopia, is a defense for the libertarian state), John Rawls (whose book A Theory of Justice outlines the philosophy of deontological liberalism, which is a system of redistributive justice), and Bruce Ackerman (who wrote Social Justice in the Liberal State, arguing for a sort of dialogic justice.)
Another important figure is John Stuart Mill who wrote Utilitarianism. Expanding on Jeremy Bentham's notion of Utilitarianism, Mill's short book influenced political calculations for over a hundred years. The main thrust of Utilitarianism is to increase the overall utility for a society. Utility is similar to happiness, so a decision or distributive scheme that would increase the societal level of happiness is better than a scheme that wouldn't--simple enough. The main critique of Bentham's Utilitarianism is the case in which there are three people and a certain distributions would give person A and B 151 utilitons each (the measurement for utility) and give Person C 0 utilitons. In other configurations, each might be able to have 100 utilitons each, however, the extra two utilitons are created by this distribution at the expense of Person C. This would create, to invoke the catchphrase, a "tyranny of the majority." Mill's Utilitarianism responded to this by adding in protections for the minority.
Some more contemporary political philosophers are Hannah Arendt, Jurgen Habermas, Ernesto Laclau, Judith Butler, Richard Rorty, and Slavoj Zizek.