Libertarianism is a political philosophy which advocates individual rights and a limited government. Libertarians believe individuals should be free to do anything they want, so long as they do not infringe upon the rights of others; they further believe that the only legitimate use of force, whether public or private, is to protect those rights. For libertarians, there are no 'positive rights[?]' to food or shelter or health care, only 'negative rights[?]' to not be assaulted, abused, robbed, censored, etc.
The term 'libertarianism', in this sense, although in itself much older, was only largely used since 1955  (http://www.daft.com/~rab/liberty/history/whois-1955.htm). The term was first introduced in the United States by thinkers who saw themselves as continuing the classical liberal tradition of the previous century. By that time the term liberalism had come to refer within the United States to belief in moderate government regulation of the economy and moderate government redistribution of wealth. These thinkers therefore called themselves libertarians; and from the United States the term has spread to the rest of the world.
However, there is still confusion, because in Europe, the French word 'libertaire', the Spanish word 'libertario', etc., of which the English term 'libertarian' is the usual translation, traditionally referred to some kind of socialist anarchism, whereas (modern US term) libertarians are not socialists at all, and most of them are not anarchists, but minarchists i.e. advocates of some minimal state -- though most of them out of pragmatism rather than out of principles.
Libertarians see their origins in the earlier 17th to 20th century tradition of classical liberalism, and often use that term as a synonym for libertarianism, particularly outside of the USA.
Some, particularly in the USA, argue that while libertarianism has much in common with the earlier tradition of classical liberalism, the latter term should be reserved for historical thinkers for clarity and accuracy. Others make the distinction to distance themselves from the socialist and welfare state connotations of the word "liberal" in American English. Still others, particularly outside the USA, use the words "libertarianism" and "classical liberalism" indifferently to denote the same tradition.
In any case, whether you equate them or not, libertarianism shares the opinions, methods, and approaches of earlier classical liberalism. It has few commonalities with so-called "welfare liberalism" or socialism. Many economically oriented libertarians use the word "socialist" nigh-interchangeably with "statist" in critiquing their opponents, even rightist opponents, out of the argument that socialism is the only consistent (family of) statist ideologies! This may perhaps be compared with Marxist use of terms such as "capitalist" and "bourgeois" in critique of other leftists (see state capitalism).
In some countries (e.g. Poland), libertarianism is called "conservative-liberalism", where "conservative" means non-socialist. In the US also, some libertarians feel conservative and some conservatives feel libertarian, because both groups recognize as theirs the ideology of the founding fathers of the USA. Still, it is possible to distinguish quite neatly two different and often opposite traditions, and it is only a matter of terminology when confusion occurs. This opposition is clearly explained in Friedrich Hayek's article "Why I Am Not a Conservative"  (http://hem.passagen.se/nicb/cons.htm).
Libertarianism has significant differences with both conservatism and liberalism (as those terms are used in the United States): see political spectrum. Libertarians consider that conservatives approve of economic freedoms but not of personal freedoms, whereas liberals approve of personal freedoms and not economic freedoms, and that they libertarians claim all these freedoms.
Libertarians prefer not to be called "right-wing." Indeed, they reject the one-dimensional left/right dichotomy and instead propose a two-dimensional space with personal freedom on one Cartesian axis and economic freedom on the other. (See  (http://www.self-gov.org/quiz).) In fact, there were times when those with libertarian views were considered left-wing on the political space (for instance, in the seventeenth century, the Whigs were revolutionaries, and in 1848, Frederic Bastiat was seating rather on the left side of the Assembly) - indeed, the balance of political opinions has shifted a lot, while the anti-political tradition of libertarianism has not moved, only evolved and grown.
The fundamental values that libertarians fight for are individual liberty, individual responsibility and individual property. Libertarians have an elaborate theory of these values that they defend, that does not always match the 'common sense' regarding liberty, and that strictly opposes collectivist views in this regard. As an example, they hold that personal liberties (such as privacy and freedom of speech) are inseparable from economic liberties (such as the freedom to trade, labor, or invest). They make this point to contrast themselves with socialists who believe that economic regulation is necessary for personal freedom, and with big-business conservatives who tie free trade with a restrictive regulation of personal issues such as sexuality and speech.
It is a chief point for many libertarians that rights vest originally in individuals and never in groups such as nations, races, religions, classes, or cultures. This conception holds it as nonsensical to say (for instance) that a wrong can be done to a class or a race in the absence of specific wrongs done to individual members of that group. It also undercuts rhetorical expressions such as "The government has the right to ...," since under this formulation "the government" has no original rights but only those duties with which it has been lawfully entrusted under the citizens' rights. Libertarianism frequently dovetails neatly therefore with strict constructionism[?] in the constitutional sense.
The classic problem in political philosophy of the legitimacy of property is essential to libertarians. Libertarians often justify individual property on the basis of self-ownership: one's right to own one's body; the results of one's own work; what one obtains from the voluntary concession of a former legitimate owner, through trade, gift or inheritance, and so forth. Ownership of disputed natural resources is more problematic and libertarian solutions such as homesteading have been studied from John Locke to Murray Rothbard.
Libertarians consider that there is an extended domain of individual freedom defined by every individual's person and private property, and that no one, neither private citizen or government, may under any circumstances, violate this boundary. Indeed, libertarians consider that no organization, including government, can have any right except those that are voluntarily delegated to it by its members -- which implies that these members must have had these rights to delegate them to begin with.
Thus, according to libertarians, taxation and regulation are at best necessary evils, and where unnecessary are simply evil. Government spending and regulations should be reduced in as much as they replace voluntary private spending with involuntary public spendings, and replace private morality with public coercion. To many libertarians, governments should not establish schools, regulate industry, commerce or agriculture, or run social welfare programs. Nor should government restrict free speech, sexual practices, gambling, drug usage, or any other 'victimless' crimes. For libertarians, government's main imperative should be "Laissez-faire" -- "Hands off!".
All libertarians agree that government should be limited to what is strictly necessary, no more, no less. But there is no consensus among them about how much government is necessary. Hence, libertarians are further divided between the minarchists and the anarcho-capitalists, which are discussed at length in specific articles.
The minarchists believe that a "minimal" or a "night-watchman" state is necessary to guarantee property rights and civil liberties, and for that purpose only. For them, the proper functions of government might include the maintenance of the courts, the police, the military, and perhaps a few other vital functions. While they are technically statists since they support the existence of a government, they would resent the connotations usually meant with this term, of trust in the government to solve any problem.
The anarcho-capitalists, believe that even in matters of justice and protection and particularly in such matters, action by competing private responsible individuals (freely organized in businesses, cooperatives, or organizations of their choice) is much better than action by monopolist governments. While they are technically anarchists, they insist in rejecting the connotations often meant with this term regarding support of a socialist utopia.
Minarchists consider that they are realist while anarcho-capitalists are utopian to believe that governments can be done wholly without. Anarcho-capitalists consider that they are realist and that minarchists are utopian to believe that a state monopoly of violence can be contained within any reasonable limits.
This division is very friendly, and not the source of any deep enmity, despite the sometimes involved theoretic arguments. Libertarians feel much more strongly about their common defense of individual liberty, responsibility and property, than about their possible minarchist vs anarchist differences. Since both minarchists and anarchists believe that existing governments are far, far too intrusive, the two factions seek change in almost exactly the same directions.
Many libertarians don't take position with regards to this division, and don't care about it. Indeed, many libertarians consider that governments exist and will exist in the foreseeable future, up to the end of their lives, so that their efforts are better spent fighting, containing and avoiding the action of governments than trying to figure out what life could or couldn't be without them. Indeed, in recent years libertarianism has attracted many "fellow-travelers" (to borrow a phrase from the Communists) who care little about such theoretic issues and merely wish to reduce the size, corruption, and intrusiveness of government.
Some libertarian philosophers argue that, properly understood, minarchism and anarcho-capitalism are not in contradiction. See Revisiting Anarchism and Government (http://www.liberalia.com/htm/tm_minarchists_anarchists.htm) by Tibor R. Machan[?].
Libertarians tend to take either one of an axiomatic natural law point of view, or a utilitarian point of view, in justifying their beliefs. Some of them (like Frederic Bastiat), claim a natural harmony between these two points of view (that would indeed be but different points of view on a same truth), and consider it irrelevant trying to establish one as truer.
An exposition of utilitarian libertarianism appears in David Friedman's book The Machinery of Freedom, which includes a chapter describing an allegedly highly libertarian culture that existed in Iceland around 800 AD.
For natural law libertarianism, see for instance Robert Nozick.
See also relevant paragraphs about this difference in points of view in the article about Anarcho-capitalism.
Libertarians do not agree on every topic. Although they share a common tradition of thinkers from centuries past to nowadays, no thinker is ever argued as a common authority whose opinions to blindly accept, only as a reference to which to compare one's opinions and arguments.
These controversies are addressed in separate articles:
In philosophy, "libertarianism" denotes an incompatibilist[?] belief which states that reality is indeterministic[?], a free action is one which is not determined, and that humans have free will (that is, that there are free human actions). In a way, this means that a free action is simply a random event.