Among all the available introductions to libertarian thought, I think Murray Rothbard's _For a New Liberty_ is the best. In it, Rothbard sets out the principles of anarchocapitalism, a system of political-economy where property rights are sacrosanct and no government exists. This is important because most libertarians support some degree of government is necessary in order to preserve a person's right to self-ownership and property.
However, Rothbard argues that the very existence of the State violates man's rights and is incompatible with freedom, even in a democratic society. This is an problem many libertarian scholars have struggled with in attempting to justify limited government. Rothbard faces no such inconsistency.
First, Rothbard introduces the concept of man's rights, establishing that the only valid right can be the right to self-ownership and ownership of one's property. With these principles -- along with the traditional libertarian non-aggression axiom -- Rothbard offers meaningful solutions to the reams of problems in today's society. He makes a forceful case that our problems would be easily solved following principles of the free market, private property, and non-aggression. Education, welfare, free speech, pollution, crime...Rothbard tackles numerous issues with great insight and clarity. In my opinion, the only significant issue he doesn't really explore is healthcare, but hey...it's a short book. (For an excellent libertarian exploration of the healthcare issue [among MANY other things], see Dr. Mary J. Ruwart's definitive _Healing Our World_. Amazon sells it.)
Rothbard introduces many ideas in this book that would be dubbed "radical" by most -- the abolishment of government police services, courts, and national defense being the most obvious. But he also believes in unlimited free speech -- this means there would be nothing illegal about blackmail or libel in a libertarian society. To most, many libertarians included, these ideas are difficult to get one's head around.
Large chapters are devoted to education, welfare, private roads, crime & private security, ecology, conservation of resources, and national defense. Some have accused Rothbard of skipping out on the private police/courts system, but this book is not a 1000-page treatise. He offers theory and historical evidence to support his ideas, but truthfully such a topic requires many books on its own. A good and short adjunct to Rothbard's ideas here is Hans-Hermann Hoppe's brilliant article "The Private Production of Defense", from the Journal of Libertarian Studies. Some good books on the issue are _The Enterprise of Law_ and _To Serve and Protect_, both by Bruce Benson, and _The Structure of Liberty_ by Randy Barnett. I believe Amazon sells all of them.
I've read this book around ten times. Worth owning if you care about where the world is headed.