I am looking for an as gentle and pedagogical as possible introduction that explains the Virasoro algebra and its applications in theoretical physics; finally I am interested in its application in string theory.

The short explanations in the physics books I am reading are not enough to make me feel comfortable about the Virasoro algebra so I need to read some more ...

I prefer shorter than whole book references which are optimally freely accessible, but if this does not work other things are welcome too.

I know a little bit about Lie algebras and conformal transformations, but the central extension issue confuses me. What I am looking for could be something at the level of these BRST lecture notes I could mostly follow.