We give a brief introduction to the study of the algebraic structures -- and their geometrical interpretations -- which arise in the BRST construction of a conformal string background. Starting from the chiral algebra \cA of a string background, we consider a number of elementary but universal operations on the chiral algebra. From these operations we deduce a certain fundamental odd Poisson structure, known as a Gerstenhaber algebra, on the BRST cohomology of \cA. For the 2D string background, the correponding G-algebra can be partially described in term of a geometrical G-algebra of the affine plane \bC^2. This paper will appear in the proceedings of {\it Strings 95}
