Question 1 - What is libertarianism and how is it related to anarchism?
The definition of libertarianism varies from person to person and also from country to country. In general, libertarianism can be defined as a philosophy that seeks to maximize individual rights and, simultaneously, minimize the power of government. Since anarchy is defined as the absence of a government, where the people have maximum individual rights, it naturally follows that anarchism is the final manifestation of libertarianism.
Question 2 - What are it's advantages/ pros?
The protection of individual rights, such as the right to protest, right to information, and others. A country with a substantial libertarian tendency can effectively keep government in check and prevent authoritarianism from ever forming.
Question 3 - What are it's cons/ critics?
While individual liberties must be preserved, it is also true that too many liberties can destroy a country. There are certain areas where a government, acting in the interest of the nation as a whole, must intervene. There is a reason why states formed in the first place; the "state of nature", with its inherent chaos and instability, in not conducive to human society. Large human societies need powerful states because we need stability.
I always found it amusing how, in the United States, a corporation is considered the same as an individual under the law, and is thus entitled to the same rights as an individual is under the Constitution. This is a blatant misuse of libertarian philosophy if I ever saw one.