I've just come back from 3 weeks of holidays and a good discussion with my parents, especially on my philosophy and libertarians.
That's a recurrent process: when I talk with a person about that I am a libertarian, the person is amused at first.
Then when that person is a socialist, I am accused of being sold to big corporations, money or whatever.
When that person is more right wing, I am disdained to be just an idealist because my view can not work.

The biggest hurdle to this kind of discussion is that most often the person is orienting the talk by bombarding me with questions on how to solve this or that "problem" without forcing people with a state.
I previously used to be trapped in this by trying to draw how people without a state would make what some want for any case such as security, education, insurance, roads or whatever.
This always leaded to a kind of brainstorm ping pong on actually designing in details such an answer to a problem currently dealt by the state.

After reading and thinking about it, I now try to make my opponent shift to a deeper and more subtle understanding of being a libertarian.
For me, the main difficulty to compare libertarians and "democrats" (not the US party but the ones that love, trust and enforce democracy) is due to that they are not aiming at the same questions.

Democrats and democracy is a question of WHAT to do WHATEVER the means, FORCE INCLUDED: the way is to let each person in the place vote and do what the most want.
More precisely, in today representative democracy, persons are voting for WHO is going to tell what to do.
That is the rule of the strongest: the state is the strongest and it is making the laws.
Once this has been done, the government has a limited period to do what it wants with any means.

Libertarianism is a question of HOW to do WHATEVER the aims: the way is to let each person in the place do what he wants except forcing people, be it as an end (like a serial killer) or as a mean (like a state).

So as a libertarian, when I want the state to stop acting, this does not necessarily mean that I don't want the aim the state was trying to reach. It means that for me NOTHING should justify to force/racket people via taxes and rules, even for apparently "heart minded reason".