I realized that the first Google hit for "conformal blocks" was a discussion at Physics.StackExchange about "A pedestrian explanation of conformal blocks".
This discussion is quite interesting and there are a number of good quality answers. But these answers were written in the span a few days, and they do not amount to a complete or satisfactory explanation of conformal blocks.
Such an explanation should probably be written as a Wikipedia article. But before writing it, one should probably rewrite the article on conformal field theory, and more generally build a decent set of articles on that subject. So, as a quick fix, I just added my own explanation of conformal blocks to the discussion in question.