Libertarians in the US want small government on three axes: they want to eliminate programs (e.g. welfare, retirement or universal healthcare), public utilities (e.g. electricity, highways), and regulation (e.g. antitrust, banking laws.) in economic terms, it's very right-wing, since it's pure unadulterated capitalism. usually they want government to "stay out of the bedroom and the boardroom" though, so they're often progressive on civil liberties. unfortunately, many self-styled "libertarians" are socially conservative, or care only about their freedoms.
Left Libertarians see both the State and Corporations as oppressive power structures, and want to reign both in. think Anarchists, but not as radical. most favor decentralized, collective government with lots of direct democracy. New Hampshire is the most right-libertarian state, while Vermont is the most left-libertarian.
the Libertarian Party in the US is ridiculously disorganized because organizing Libertarians is like herding cats. afaik there aren't really unified Libertarian parties anywhere in the world, though maybe e.g. the Pirate Party would be close?
critiquing both is what makes you a left libertarian.