The principles of liberty
The most fundamental premise of libertarian thought is: "You own your life."
In other words: Freedom, liberty. Unless you're ready to give up on America, those words should still have some meaning for you.
This doesn't mean freedom at any cost, or the liberty to hurt people; for now, let's say it simply means the freedom to pursue happiness.
The political direction of libertarians tends to run toward the economically conservative and the socially liberal. Both of these promote basic freedom by decentralizing monetary power and the associated control and by acknowledging the freedom of each person to choose his or her best life.
Perhaps you think that other people should do one thing or not do another because of the values you hold. In all frankness, it is not your responsibility or privilege to dictate anyone else's pursuit of happiness, any more than I should be allowed to say whom you have to marry or what kind of car you have to drive.
Conservatives tend to think that it is necessary to tell people how to behave. Liberals tend to think that it is necessary to tell people how to get along with each other. Libertarians hold a basic respect for all people, their choices, and their beliefs, regardless of their abilities or position. Only when one perceives himself as above or better does the guardian mentality enter into politics to condone forcing people into a particular behavior or away from a behavior that does no harm to anyone else.
Libertarians recognize that we all stand together in true equality of spirit and hope.
The biggest fallacy about libertarians is the incorrect assumption that an objection to the use of force for a certain thing is the same as an objection to the thing itself.
For example, take the issue of community charity. Giving to one's community is a good and worthy thing, but if I say you have no right to force me to give, you might easily assume that I am against charity itself.
You can also see how the fallacy applies to undesirable elements. Take the issue of drug use. Using drugs can be a terrible, debilitating thing, but if I say you have no right to force me not to use drugs as long as my using does not hurt you, you might easily assume that I endorse the use of drugs.
In both cases the objection is to the use of force to impose one set of values over another.
This distinction is often lost. Libertarians say, "Don't force me to pay for your social programs," and people hear, "I don't care about anybody but myself." Libertarians say, "Don't tell me what drugs I can or can't take," and people hear, "Drugs are great, give them to your kids."
This doesn't mean that if we're all left to pursue our own best life we will magically make the best choices. Sometimes people and companies will lie and cheat and hurt. The primary (but not the only) role of government, therefore, is to provide courts, enforcement, and punishment against these things.
Government, according to libertarians, should be reactive and not proactive. For example, can you write down all the ways in which children on a playground might hurt each other? What would a proactive playground law look like? Rather than trying to enumerate all the dangers, from throwing grasshoppers to peeing on the swing to fake swordfights with twigs, it's much better to be guided by more basic pinciples: be nice, don't hurt, don't lie, and deal with the trouble makers when they make trouble.
The world is not exactly like a children's playground, but the same principle applies: if you try to proscribe all the what-ifs, it's hard to find a place to stop, and the end is a loss of common sense, a loss of everyday responsibility, and a world in which you can be punished for simply having the capability to do wrong.
Libertarian policies are based on the best common sense of what works in a community of free people.