Libertarian is also known as Classical Liberalism, Randian/Randianism (after Ayn Rand), but is just the belief that liberty should be the core principle of their philosophy, seeking to maximize political freedom and autonomy, emphasizing freedom of choice, voluntary association and individual judgment (and responsibility/accountability). In Europe, this would be left wing, in the U.S. it is right wing.
The basics are Thomas Jefferson was a libertarian, as were most of the founding fathers. Which means most libertarians are conservatives, in that they want to go back to when the U.S. had more individual liberties, smaller federal government (more state governmental control), lower taxes, more opportunities, and less regulations. There's a huge spectrum of how far back, and in what areas first. And it's important to remember that this is opposed to most European conservatives: where when they want to go back, it's often back to more imperial or monarchistic autocracy or centralized control (our histories are different, so back means different things).
Also, libertarianism is a philosophy that can't always be achieved. (At least to the more pragmatic/realistic). So it's something to be strived towards, with gaps. Jefferson felt we shouldn't go into national debt... then the Louisiana purchase opportunity came up, and he said, "screw that, I'm buying the land rights!". He also had slaves, while hating the institution and wanted to figure a way out of the economic trap it was for both sides. Things like that.
Libertarians are "classical liberals". Basically, what liberal (where libertarian came from) originally meant is, "Liberty": maximizing autonomy, freedom of choice, and the primacy of individual judgment. Around the turn of the 19th century, American progressive socialists started to usurp the name liberal, and re-invent it as meaning themselves (heavy state control and what federalism used to mean): even-though everything the left stood for was the opposite of classical liberalism. Since this didn't take root in Europe, "liberal" means opposite things in the U.S. and Europe (generally American libertarian is what Europe means with liberal -- though Americas misuse of the term has muddied the waters as Europeans have started using it wrong as well).
Here's a list of things that are, or are not, Libertarian -- and what the term or philosophy means.
Am I a libertarian? - While I lean little-l libertarian, I'm not really a real Libertarian (or that's what they tell me). I'm for less government (especially federal), for more isolationism and less militarism, and for more individual liberty and property rights and so on -- all things that align well with Libertarianism. But I also believe things that contradict most people that call themselves Libertarian (though isn't a requirement of the dogma, seems to be canon, for most). Things like I believe that society has a speed limit on change, or you get instability -- so not only which direction you go, but how quickly matters. I'm for some military intervention, I believe in fair trade over free trade (the market will work out free trade in long run, but in the short run people get hurt -- so you need limits), I'm for some social services (just less), and believe that most things (like the legalization of drugs) must come very slowly and with education or there will be backlash. Thus, I call myself more a conservatarian -- conservative on how quickly I'd want libertarianism, and conservative on how far down that path I'd go.
Big-L versus Little-l - One quick aside, libertarianism (with a little-l) is a philosophy or belief system, with many branches, while big-L Libertarian is a political party. Just like Republicans and Democrats, no two Libertarians can seem to agree on what the priorities should be, and how to implement every tenet of their philosophy. Since quirks of our language allow for mis-capitalization, a lot use big-L when talking about little-l, so it always helps to clarify whether someone who says libertarian means philosophy or party.
Left Libertarian - Left-Libertarian is an oxymoron, like kind rapist or practicing compassionate gennoacide. You are either trying to increase individual liberty (and accountability) and are on the libertarian spectrum, or you're trying to seize the means of production and force social equality through government force, but not both. You can't pretend to be about individualism, then say that's why you want to use the state to force collective values. So anyone that says Left-Libertarian without openly mocking the hypocrisy of it, lets you know that they're just a Social Justice Warrior trying to wear the costume of a freedom loving person, but they aren't.
- While walking home one evening I came upon a man standing at the edge of a bridge, about to jump. “Wait!” I yelled, and ran over to see what was the matter.
- “It’s this country,” he lamented. “It’s falling into ruin and there’s nothing I can do about it. I don’t want to live on this planet anymore.”
- “Cheer up,” I said. “We’re all in this together. Say, are you a liberal or a conservative?”
- “I’m a libertarian,” he said.
- “Me too!” I exclaimed. “See, you’re not alone. Are you a free-market libertarian or a libertarian socialist?”
- “Free-market libertarian,” he said.
- “Same here!” I said. “Paleo-libertarian or neo-libertarian?”
- “Paleo-libertarian,” he said.
- “Hey, so am I! Chicago or Austrian school of economics?”
- “Austrian,” he said.
- “Me too! Hayekian or Rothbardian?”
- “Rothbardian,” he said.
- “Same as me! Consequentialist or deontological?”
- “Consequentialist,” he said.
- “Die, statist!” I yelled as I pushed him off the bridge....
Libertarians are anarchists - There is a reductio ad absurdum argument by the left, that Libertarians are all anarchists, or it would lead to anarchy. This is like saying anyone for a single social program wants complete Authoritarian Marxist Tyranny. There might be a few that far out on the bell curve, but they're not the norm of the movement, nor close. This also devolves into idiotry about "who would build the roads, schools, police". We had all of those back before we had an IRS and were far more Libertarian than most Libertarians today would want -- so it's an argument that appeals to the ignorant or partisan, not the informed. We can debate how extreme a Libertarian is, and whether they are an anarchist in Libertarian clothing (and there are a handful out there). But saying Libertarians are anarchists is like calling all Democrats rapists, just because Bill Clinton was one.