The biggest form of Libertarians is in the American Libertarian party which formed back in 1971 after some minor parties and groups with similar views. It stands today as America's biggest 3rd party, even though they haven't elected a single major politician that I know of under the Libertarian banner though some have either been Republicans or still run as Republicans.
The main three slogans are Free Will, Free Markets, and Personal Responsibility. Which is plainly translated as no more governmental laws, and no more welfare of any kind.
Is welfare such a bad idea? Back before social security, old people had to rely heavily on their family members, or fraternal groups just to keep on living. Ask how easy it is for old people to get private health insurance or how they'll be able to pay for it. Hope you saved up your money before you lose your job too, because you wouldn't have unemployment to fall back on either. Free school lunches for kids? That's communism to Libertarianism.
Libertarianism and anarchy go hand in hand many times. The term was first used by French Anarchistic-Communist Joseph Déjacque who needed a new term to avoid anti-anarchy French officials. Some Libertarians will say they are not anarchists, some will admit it, and some will say they feel part anarchist. But they will at least admit to having an army to make sure the commies don't invade us, and a police to make sure them hoodlums don't get their nice things.
Is Libertarianism left or right? Well it's often considered left wing-right wing politics. Meaning they are not on either side. They are usually left socially and on the far right fiscally. I can respect their liberal social views, but when it comes to their anarcho-capitalism I can't so much. We are ruled more by corporate CEOs than the government these days. I know what's it's like working for people like that, and that's about as close as slavery as you can get for entry-level workers. People lose their morals when they try to make more money. BP tried to save money by avoiding equipment costs, and then their tank blew up and leaked oil everywhere and destroyed a lot of the marine life and economy. Thanks BP!
Now Libertarians can usually champion any idea they have since most of the views haven't existed in the modern world for ages, and no pure Libertarians society has ever existed. But if they have a idol, that most defiantly is Republican Congressman of Texas, Ron Paul. A private doctor who's made millions and invested in gold mine would never vote for universal health care, but will vote to go back the gold standard. I wonder why?
Libertarianism may have some good ideals, but I think their only use is to help balance out a government that might lean a little too left or too right on social or fiscal issues. I think balance is key, not an extremist reversal on issues.