Libertarianism

Simple

Libertarians are socially liberal and economically conservative. Social freedoms can be represented by the bedroom and economic freedoms can be represented by the boardroom.

• Libertarians = keep government out of both the bedroom and boardroom 
• Liberals = keep government out of the bedroom but in the boardroom 
• Conservatives = keep government in the bedroom but out of the boardroom 
• ??? = keep government in both the bedroom and the boardroom. 

Another simple way of thinking about libertarianism is that, to paraphrase Oliver Wendell Holmes Jr... the freedom to swing your fist ends where somebody else's nose begins. This concept is specifically referred to as J.S. Mill's Harm Principle.

Less Simple

Libertarianism is an ideology of limited government. Why should the government be limited? There are two main theoretical justifications...consequentialist (results) and deontological (rights).

The consequentialist theory is that, for most goods, the invisible hand is more efficient at allocating resources than the visible hand (planners - congress). Therefore, the state should be limited to producing goods that society values but, due to lack of financial incentive, the market largely fails to produce.

The deontological theory is that property rights trump "most" market failures. The exceptions are those public goods that deal with protecting your property. Therefore, the state should be limited to providing goods that help protect your property from others.

Practical

Libertarians want to move welfare, education, healthcare and numerous other public goods over to the private sector. These goods can be thought of as Pseudo Public Goods (PPGs) because they only loosely match the economic criteria of a public good. From the consequentialist perspective it's generally assumed/believed that the private sector will produce comparable (if not greater) levels of PPGs as the public sector. From the deontological perspective, given that property rights trump PPGs, it's irrelevant whether the private sector will produce the same levels of PPGs as the public sector.

Criticism

The free-rider problem is by far the strongest argument for not moving PPGs over to the private sector. If people had a choice whether or not to fund PPGs it's very likely that funding for PPGs would decrease significantly.

On the off-chance you manage to convince a libertarian that the private sector will under-produce PPGs they can always fall back on the deontological argument. It's their ace in the hole.

Trying to attack the deontological justification for libertarianism is as useless as trying to attack the 10 Commandments. Everybody has different value systems...they are all subjective and as such cannot be proved wrong.

Recently though I was pleasantly surprised when a member of the Libertarian Party on facebook shared a link to Jeffrey Miron's blog where he acknowledges that the free-rider problem isn't just applicable to national defense...Libertarianism and Anti-Poverty Programs *

Solutions

Libertarians, liberals and conservatives all have different value systems. It's fairly reasonable that people do not have the choice whether they pay taxes but it's very unreasonable that people are forced to fund public goods that they do not value. Tax payers should have the liberty to use their tax dollars to directly fund the public goods that they believe government should be responsible for providing. This is the pragmatarian approach.

The pragmatarian approach would allow the invisible hand to decide whether the public or private sector is better at producing a "public" good. The result would be...

1. an optimal division of labor between the private and public sector
2. public goods would be produced with maximum efficiency
3. levels of all public goods would accurately reflect society's values
4. everybody would have more freedom
5. government would have less power
6. taxpayers would transform into donors...and they would feel a "warm glow" when they made their "donations" to public goods.

The question then becomes...is it easier for liberals or libertarians to fully appreciate the pragmatarian solution? Pragmatarianism provides more freedom then we currently have...so libertarians should easily recognize it as being a far superior option when compared to the current system. However, some might refuse to concede that the free-rider problem is a legitimate problem. Liberals on the other hand might not sufficiently "grasp" the invisible hand to realize how much benefit they would derive from the pragmatarian approach.

So in other words, we can say that pragmatarians are either libertarians that understand the free-rider problem or liberals who understand the invisible hand.

To conclude here are a bunch of quotes. The first set establishes the connection between classical liberalism and libertarianism. The second set establishes who the most prominent libertarians were/are and the last set establishes what they thought/think about the scope of government.

Classical Liberalism and Libertarianism

• The word liberal has an odd history. It comes from the same root as liberty, and originally it simply meant "free." In that broad sense, the United States as a whole is a liberal country, and all popular American ideologies are variants of liberalism. In a more restricted definition, a liberal was a person who believed in limited government and who opposed religion in politics. A hundred years ago, liberalism referred to a philosophy that in some ways resembeled modern-day libertarianism. For that reason, many libertarians today refer to themselves as classical liberals. - American Government and Politics Today 2008: The Essentials
• The classical liberal movements of the eighteenth and nineteenth centuries are, of course, the forebears of contemporary libertarian thought. - Total freedom: toward a dialectical libertarianism
• Contemporary libertarians believe, with some legitimacy, that the greatest threat to liberty is an expanding government with a monopoly on state power. Their answer: limit government, protect a basic skein of fundamental rights, and the rest will work itself out. In this respect, libertarians are true heirs of the classical liberal tradition. - The Political Centrist
• Libertarianism has been more an intellectual than popular movement since its ancestor, classical liberalism, was first articulated by John Locke. William Maddox and Stuart Lillie identified six tenets of classical liberalism to which American libertarians subscribe in a modified form today: individualism, an instrumental view of the state, limited government, individual rights, legal equality, and representative government. These six tenets cluster around two domestic policy questions - the proper role of government and the prescriptions for apparent social inequalities. - Hostile takeover: the House Republican Party, 1980-1995
• Scalet and Schmidtz quite correctly identify the classical liberal contribution to political theory as its focus on limited government, rather than (as classical liberals are often accused of preferring) "weak" government. The size of government is not the primary concern of classical liberals; its limits are. Limited governments tend to be small relative to unlimited governments. They also note that "classical liberals have been champions of democracy." The two issues - limited government and democracy - have traditionally been linked together in classical liberal thought by the theory of constitutionalism, which limits the powers of majorities no less than of minorities. - Realizing freedom: libertarian theory, history, and practice

Most Prominent/Influential Libertarians

This source establishes the prominence of Jefferson, Tocqueville, Adam Smith, John Stuart Mill, Ludwig von Mises, Friedrich von Hayek and Ayn Rand...

Libertarians cite as progenitors Jefferson, Tocqueville, Adam Smith, and John Stuart Mill as well as economists of the Austrian school Ludwig von Mises and Friedrich von Hayek, author of The Road to Serfdom (1944). But most Republican libertarians were first inspired not by these classical liberals but by the author of The Fountainhead and Atlas Shrugged. - New York Magazine Mar 4, 1996

This source establishes the prominence of John Stuart Mill and Herbert Spencer...

John Stuart Mill, the great and generous theorist of liberalism, and Herbert Spencer, a major exponent of laissez-faire individualism, whose writings appealed immensely to the Spanish anarchists, can be - and have been - rightly designated as 'libertarians' - Anarchist seeds beneath the snow

This source establishes the prominence of John Stuart Mill...

In contemporary times, libertarians have positioned themselves as the heirs to J. S. Mill and his defense of individual liberty. Their ideas have grown increasingly influential. Contemporary libertarians embrace Mill's On Liberty because it "sounds important libertarian themes: that individuals should be free to live as they choose so long as they don't harm others and that the power of government should be strictly limited." - Putting ideas to work: a practical introduction to political thought

This source establishes the prominence of Robert Nozick...

With thirty years' distance on its publication, one can safely assert that Robert Nozick's Anarchy, State, and Utopia (1974) has achieved the status of a classic. It is not only the central text for all contemporary academic discussions of libertarianism; together with John Rawls's A Theory of Justice (1971), it arguably framed the landscape of academic political philosophy in the last decades of the twentieth century. - Natural rights liberalism from Locke to Nozick, Part 1

This source establishes the prominence of John Stuart Mill, Ludwig von Mises, Friedrich Hayek and Ayn Rand...

This political philosophy has a long pedigree. It has roots in the classical liberalism of eighteenth- and nineteenth- century British thinkers such as John Stuart Mill, and, more recently, in the Austrian school of economics represented more powerfully in the United States by Ludwig von Mises and Friedrich Hayek. On a more popular level, the novelists Ayn Rand and Robert Heinlein proved effective propagators of a radical anti-statist, individualist ethos, and according to historian Brian Doherty, more than half of the earliest Libertarians cited one of those two authors as their primary ideological influence. - Encyclopedia of American political history

Libertarians on the Scope of Government

1776 - Adam Smith (Michael Howard)

Although Smith was against governmental interference with the market, he had a a theory of government sometimes known as the "duties of the sovereign". The system of natural liberty required the sovereign to perform three duties; defense, the exact administration of justice, and the erection and maintenance of public works. Even though he was a libertarian, Smith realized that the market could not provide certain public goods which were too expensive for provision by private individuals. - Public Sector Economics for Developing Countries

1848 - Frederic Bastiat

For ourselves, we consider that Government is and ought to be nothing whatever but the united power of the people, organized, not to be an instrument of oppression and mutual plunder among citizens; but, on the the contrary, to secure to every one his own, and to cause justice and security to reign. - Government

1865 - John Stuart Mill

That the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. - On Liberty

1865 - Herbert Spencer

And now mark that whether we consider government from this point of view, or from that previously occupied, our conclusions respecting it are in essence identical. For when government fulfils the function here assigned it, of retaining men in the circumstances to which they are to be adapted, it fulfils the function which we on other grounds assigned it — that of protector. To administer justice, — to mount guard over men's rights, — is simply to render society possible. And seeing that the two definitions are thus at root the same, we shall be prepared for the fact that, in whichever way we specify its duty, the State cannot exceed that duty without defeating itself. For, if regarded as a protector, we find that the moment it does anything more than protect, it becomes an aggressor instead of a protector; and, if regarded as a help to adaptation, we find that when it does anything more than sustain the social state, it retards adaptation instead of hastening it. - Social statics

1884 - Herbert Spencer (Tom G. Palmer)

The challenge facing Americans today in defending constitutionally limited government was succinctly stated by the English libertarian Herbert Spencer in 1884: "The function of Liberalism in the past was that of putting a limit to the powers of kings. The function of true Liberalism in the future will be that of putting a limit to the powers of Parliaments." - Realizing freedom: libertarian theory, history, and practice

1944 - Ludwig von Mises

Liberalism differs radically from anarchism. It has nothing in common with the absurd illusions of the anarchists. We must emphasize this point because etatists sometimes try to discover a similarity. Liberalism is not so foolish as to aim at the abolition of the state. Liberals fully recognize that no social coöperation and no civilization could exist without some amount of compulsion and coercion. It is the task of government to protect the social system against the attacks of those who plan actions detrimental to its maintenance and operation. - Omnipotent Government

1962 - Milton Friedman

First, the scope of government must be limited. Its major functions must be to protect our freedom both from the enemies outside our gates and from our fellow-citizens: to preserve law and order, to enforce private contracts, to foster competitive markets. Beyond this major function, government may enable us at times to accomplish jointly what we would find it more difficult or expensive to accomplish severally. However, any such use of government is fraught with danger. We should not and cannot avoid using government this way. But there should be a clear and large balance of advantages before we do. By relying primarily on voluntary co-operation and private enterprise, in both economic and other activities, we can insure that the private sector is a check on the powers of the governmental sector and an effective protection of freedom of speech, of religion, and of thought. - Capitalism and freedom

1963 - Ayn Rand

The proper functions of a government fall into three broad categories, all of them involving the issues of physical force and the protection of men’s rights: the police, to protect men from criminals—the armed services, to protect men from foreign invaders—the law courts, to settle disputes among men according to objective laws. - The Nature of Government

1973 - Friedrich Hayek (Alan O. Ebenstein)

These words were even more significant because of the government services to which he applied them - "without exception to all those services of which government possesses a legal monopoly, with the only exception of maintaining and enforcing the law and maintaining for this purpose an armed force, i.e. all those from education to transport and communications, including post, telegraph, telephone and broadcasting services, all the so-called 'public utilities,' the various 'social' insurances and, above all, the issue of money." In the last pages of Law, Legislation and Liberty, published in 1979, Hayek the classical liberal became Hayek the libertarian. - Friedrich Hayek: a biography

1974 - Robert Nozick

Our main conclusions about the state are that a minimal state, limited to the narrow functions of protection against force, theft, fraud, enforcement of contracts, and so on, is justified; that any more extensive state will violate persons' rights not to be forced to do certain things, and is unjustified; and that the minimal state is inspiring as well as right. Two noteworthy implications are that the state may not use its coercive apparatus for the purpose of getting some citizens to aid others, or in order to prohibit activities to people for their own good or protection. - The Political Theory of Robert Nozick

1995 - Richard A. Epstein

This basic insight - law must control the most lawless - lies behind the strong insistence on the basic rules of ordinary society. It also explains the libertarian's constant theoretical emphasis that the function of government is to control the use of force and fraud against the person and property of others. - Forbidden Grounds: The Case Against Employment Discrimination Laws

1996 - Dick Armey (Jacob Weisberg)

While one can reject this notion of a stripped-down state, libertarianism is a principled and coherent worldview. It provides an answer to every question. Police departments and the army - yes. Just about everything else - no. Ask most politicians, from Gingrich to Clinton, what the role of the federal government is, and you'll get a stream of mush. Poke a libertarian and you'll get a response like the one Dick Armey gave shortly after becoming majority leader: "Defend our shores, build a system of justice, and construct some infrastructure. Gee, I'm running out of other suggestions." - New York Magazine Mar 4, 1996

1998 - David Boaz

Libertarians argue that we can and should move a long way toward minimal government; outside of the protection of our rights by police, courts, and national defense, it's hard to think of goods and services that could be produced more efficiently by a government bureaucracy than in the competitive marketplace. Libertarianism: A Primer

2004 - James Walsh

Libertarians accept the need for a limited state - if only to provide basic levels of safety and security. Their focus is keeping the state limited to a disciplined - and small - number of activities. Anarchists still want to smash the mechanisms of state. As I've noted, anarchy is an emotional system. - Liberty in Troubled Times: A Libertarian Guide to Laws, Politics and Society in a Terrorized World

*[Update - 29 Oct 2011]