Libertarianism is an ideology that believes in a small government and generally opposes government intervention in the economy and in personal life.
While there are some similarities to conservatism, libertarianism is far more sympathetic to a laissez faire approach to social issues (e.g. pro-gun, pro-gay marriage, etc.) whereas conservatives are more sympathetic to supporting traditional social norms or values.
There are several sub-branches of libertarianism. One branch is highly skeptical of all government intervention and believes in a minimal “night watchman” state (whereby the only function of the government is to provide a minimum level of national defense and security). Many adherents to this school of thought are highly hostile to the government, believe taxation is theft and strongly support robust gun rights. Some academics who have influenced this movement include Robert Nozick (early work).
A second wing, more closely associated with economists like Ayn Rand, Bryan Caplan or the public choice school of thought focuses more on the economic costs of government action and is less overtly hostile to the existence of the government. They focus less on the principled wrong of government action and more on the practical benefits of free association, free markets and free movement. They care more strongly about deregulation and privatization. Some issues they’d support include abolishing the FDA and FDIC, voucherizing public schools and lowering taxes.
A third wing, more closely associated with economists like Tyler Cowen, is often called “state capacity libertarianism” (with a related concept of “liberaltarianism”). This school of thought believes that many or most government interventions are counterproductive and markets are a socially valuable way of allocating resources in society, yet also believe a competent government is necessary to protect robust markets and individual rights.