American Libertarianism is mish-mashed together from anarcho-capitalists (who still believe a state should exist to defend property rights) Objectivists (both the ARI and the non-tool kind) and civil libertarian/capitalist types. In general, we’re talking about civil libertarians who believe strongly in property rights.
Anarchism/Libertarianism in Europe generally means anarcho-syndicalism, that is that the state should not exist to defend property rights. This isn’t to say there would be no private property, but certainly no private ownership of land or the means of production.
In some circles, “Libertarianism” refers only to civil liberterarianism to dodge confusion over the property rights issue.
I think the idea is that humans have passed beyond mere primate-hood and can be responsible and work together for the common good without anyone holding a whip. No, I don’t really buy it either. It certainly couldn’t work in America.
Libertarians would like to believe the same thing, but we realize that there are some things (and some people) that simply cannot be handled as effectively without a good proxy (ie, the government). We think the government is useful, true, but only as far as it can be controlled, much like a car is useful only as far as it can be made not to crash into things or run people over.
In other words, the government is a tool of the people as a whole. Referenda and elections make it accountable to the majority, and a constitution makes it harmless to the minorities.
Myself, I think people are generally good and responsible, if only because we like to avoid hassle. Ergo, I think a massive and powerful government is akin to putting a docile dog in a reinforced concrete bunker: it’s both unnecessary and cruel.