It seems like there are already a lot of good responses in this thread, but let me see if there is anything that I can add through my response:
Libertarianism is a political philosophy that is an aggregate of classical liberalism and free-market capitalism. It attempts to yield the fruits of freedom and peace by putting a defense of private property at the forefront. In the words of the libertarian political philosopher and economist Ludwig von Mises, [libertarianism] can be described as the program of freedom, peace, and property.
Many libertarians agree on this much. The conclusions that this political philosophy can lead to, however, leads to a great diversity of thought within libertarianism. To borrow from the free-market economist Walter Block, there are four types of libertarianism: anarcho-capitalism, minarchism, constitutionalism, and classical liberalism.
Anarcho-Capitalism is the trifecta of anti-war, anti-State, and pro free-markets. There are different arguments in favor of anarcho-capitalism, ranging from the deontological argument based on the Non-Aggression Principle to consequentialist arguments that empirically demonstrate anarcho-capitalism to produce the greatest utility for the greatest amount. Scholars under this school of thought include Murray Rothbard, David Friedman, and Michael Huemer. They all present different arguments in favor of anarcho-capitalism and I think it is beneficial getting acquainted with all of them.
Minarchism is, in the words of Robert Nozick, the nightwatchman-State. This school of thought suggests that the only legitimate function of government is to defend the individual from force, fraud, and theft. This system, from which there are also many different forms of arguments in favor of it, espouses minimal taxation as necessary to fund a military, police departments, and courts. Other than these functions and institutions, the government ought to do no more. Some of the leading scholars from this school of thought might include Ludwig von Mises and Robert Nozick.
Constitutionalists are those who advocate for government to operate strictly under the confines of the Constitution. If this were to be achieved, most of #2 would be achieved in addition to some extra government functions that are the result of interpretations of the Constitution by the Supreme Court. Ron Paul is pretty well known for being a strong Constitutionalist, although it is clear from Dr. Paul's books that he is more in line with #2 and sympathizes with #1.
Classical liberalism, as Block uses it here, describes those who advocate for a nightwatchman-State with additional functionality. The additional functions would include social safety nets, universal healthcare, a voucher system for schools, and certain central banking controls until central banking can be abolished (increasing the money supply at a steady rate). Some of the most well-known academics under this school would include Friedrich Hayek and Milton Friedman.
*It is crucial to point out that I use the word [classical]-liberalism as Mises would, a synonym to the phrase "freedom, peace, and property." This might come into clash with Dr. Block's use of the word when describing the ideas of Hayek and Milton Friedman. Nonetheless, I think that Block's classification is really helpful so I've opted to leave the original language. I just wanted to avoid confusion, because I do not mean to say that libertarianism is a blend of "universal healthcare" and private property.
I think that each school of thought is worth studying, because at the end of the day each school has similar goals in mind which in summation create what is known as libertarianism.
If I had to summarize my entire explanation with one word, I would choose: property.