Lets $M_A$ be the space of metrics of area $A$ on a two dimensional surface $S$, and let $D_0$ be the group of area-preserving diffeomorphisms whose right action on $M_A$ is given by pullback. The author says that $\frac{M_A}{D_0}$ is the space of geometric structures of area $A$ in a given conformal class.
What exactly does the last sentence mean? What is a geometric structure and what does it mean for one to be in a conformal class of another? Why is this quotient non-trivial? Is it not true that given two metrics of the same area that there is always a diffeomorphism that pulls back one metric to the other?