What does it mean when one states that a ring structure varies continuously with some variable? Say, in the following context
The stalks may be rings, and in that case the ring structure of $\mathscr{F}_x$ is assumed to vary continuously with $x$. Or, each $\mathscr{F}_x$ may be a module over some fixed field.
taken from Zariski's Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory.
A sheaf of sets on a space $X$ is a topological space $\mathscr{F}$ together with a continuous map $p:\mathscr{F}\to X$ satisfying certain conditions (this is not the only way to define a sheaf, but it is the one Zariski is using). If you put a ring structure on $\mathscr{F}_x=p^{-1}(\{x\})$ for each $x$, then to say this ring structure "varies continuously" just means that the ring operations are continuous maps (and similarly for any other sort of algebraic structure). For instance, addition is a map $+:\{(a,b)\in\mathscr{F}\times\mathscr{F}:p(a)=p(b)\}\to\mathscr{F}$, and this map should be continuous (with respect to the product topology on the domain). Similarly, multiplication should be continuous, negation should be continuous as a map $\mathscr{F}\to\mathscr{F}$, and the maps $0,1:X\to\mathscr{F}$ which send $x$ to the zero element or the unit element of $\mathscr{F}_x$ (respectively) should be continuous.
(The more common modern approach is to define a sheaf not in terms of the space $\mathscr{F}$ but rather in terms of the operation which takes an open set $U\subseteq X$ to the set $\mathscr{F}(U)$ of sections of the map $p$ over the set $U$, together with restriction maps $\mathscr{F}(U)\to\mathscr{F}(V)$ whenever $V\subseteq U$. In this framework, it is automatic that any algebraic structure on the sets $\mathscr{F}(U)$ "varies continuously", in the sense that putting (for instance) a ring structure on each $\mathscr{F}(U)$ such that the restriction maps are homomorphisms is equivalent to putting a ring structure on each stalk $\mathscr{F}_x$ such that the ring operations are continuous as described above.)
Suppose you have a vector bundle on a manifold, in such a way that each fiber is a ring. Consider a local trivialization. In terms of it, you can describe the multiplication of each of those rings in terms of structure constants: that the structures of rings varies continuously means that those structure constants are continuous functions.
In other categories, like the holomorphic one or the algebraic one you adapt as needed.