How do you define capitalism?

I define it as a system of private property and voluntary trade. Private ownership of the means of production is accurate too.

How do you conciliate AnCap theory with market failures.

I don't think there's a perfect political philosophy out there. From a consequentialist point of view I think the cost of the market failures of capitalism are far smaller than the cost of government and it's failures. I'm also not entirely sure what you mean by market failures, so I'm giving a vague answer. Here's a lecture by David Friedman where he goes over this exact subject.

Is an AnCap any different from (American) libertarianism? If so, how?

Well, I don't think Anarcho-Capitalism or libertarianism is tied to one country. The source of a lot of libertarian theory are people like Bastiat (French), Mises (Austrian), Hayek (Austrian). With the internet like it is, I also think the divisions of countries with regards to ideology is washing away.

That being said, the answer yes, and no. You could think of an AnCap as a consistent libertarian. We take the philosophy of libertarianism and we apply it without exceptions. The non-aggression principle is a nice summary of that philosophy. We all apply this principle in our daily lives, and we apply it to governments too.

For a more Brazilian problem, how would AnCap solve the issue of expensive/unaffordable quality education for the majority of the population?

I'm not familiar with the situation in Brazil, but I answered a similar question yesterday:

Listen to this interview. James Tooley, a professor of education policy, originally opposed to privatization started researching the subject after stumbling onto cheap private schools in India (It's been five months since I listened, so I'm not sure I'm remembering everything correctly). He found similar schools all over the world, especially in poor countries, but also in countries with public schools, where people chose the private schools because they were more convenient, and you know, not shit. Basically, private schools are already educating the poorest of the poor, today. It's not really that much of a theoretical question.

Generally though, the voluntary and private production of anything is cheaper and more efficient than nationalized alternatives, where you end up with shortages and inefficiencies that leads to horrors, like millions or dead of starvation, or the 1975 trabant.