I'm trying to read "Introduction to Algebraic Geometry and Algebraic Groups" by Demezure and Gabriel and I'm already stuck on the following definition. A geometric space is defined to be a pair $(X, \mathcal O_X)$, where $X$ is a topological space and $\mathcal O_X$ is a sheaf of rings, such that for each $x \in X$, the stalk $\mathcal O_{X,x}$ is a local ring with unique maximal ideal $\mathfrak m_x$.
Definition (word for word): A morphism of geometric spaces $f: (X, \mathcal O_X) \rightarrow (Y, \mathcal O_Y)$ consists of a continuous map $f^e:X \rightarrow Y$ and a homomorphism of sheaves of rings $f^f$ of $\mathcal O_Y$ into the direct image $f(\mathcal O_X)$ of $\mathcal O_X$, such that for each $x \in X$, the homomorphism $f_x: \mathcal O_{Y, f(x)} \rightarrow \mathcal O_{X,x}$ induced by $f^f$ is local, i.e. satisfies $f_x(\mathfrak m_y) \subseteq \mathfrak m_x$.
What is meant by a "homomorphism of sheaves of rings?" Is this a natural transformation between the sheaves? (they are contravariant functors) What is a direct image? How does $f^f$ induce a homomorphism between stalks? Finally, what is $y$?