You can think of the points of $\mathrm{Bun}_G(X)$ as the set of maps $$X\longrightarrow \mathrm{B}G$$
where the $\mathrm B G$ is the classifying "space" (stack) for principal $G$-bundles in the algebraic category (i.e. it classifies algebraic principal $G$-bundles over schemes) or, if you don't like schemes, you can do this story in the category of analytic spaces, as we will do in the following paragraphs.
Actually, in the formalism of stacks, the points of $\mathrm B G$ do not form just a set of points but a groupoid: specifically, you think of $\mathrm B G$ as the action groupoid associated to the trivial action of $G$ on the point $*$. That's a perfectly legitimate groupoid internal to the category of analytic spaces, which means its set of objects and set of morphisms are actually analytic spaces - even complex manifolds in this case. You can also think of $X$ as a groupoid: the action groupoid of the trivial group acting on $X$. Since $X$ and $G$ are nice smooth complex manifolds, you don't even really need to think about non-reduced analytic spaces until you need to describe infinitesimal properties of $\mathrm{Bun}_G(X)$ (for example, maps from non-reduced irreducible zero-dimensional spaces -"fat points"- are enough to determine if $\mathrm{Bun}_G(X)$ is smooth).
Maps $X\to \mathrm B G$ are just maps of (analytic) groupoids, and they form themselves a (set theoretic) groupoid.
To really "know" what the stack $\mathcal M$ (e.g. $\mathrm{Bun}_G(X)$) "is", it's not enough to know its points: you also have to know which are the morphisms to it and from it involving another space or, more generally, groupoid (actually, by Yoneda's lemma it's enough to know the morphisms to it from spaces). Morphisms involving $\mathcal M$ to or from another analytic groupoid $\mathcal N$, are by definition Morita morphisms of groupoids. More succintly: the category of analytic stacks is the category with objects analytic groupoids and morphisms Morita morphisms of analytic groupoids. So, in particular, Morita equivalent groupoids are to be thought of as isomorphic stacks.
(There is a little sublety though: what I called the "category" of analytic stacks is actually a $2$-category in that maps of stacks are to be considered somehow "homotopic" when there is a $2$-morphism between them, pretty much in the same way as two functors between categories are to be regarded as "homotopic" if there is a natural isomorphism between them).
For $\mathrm{Bun}_G(X)$ there is another, equivalent but probably more expressive, characterization (or definition?) of morphisms $S\rightarrow\mathrm{Bun}_G(X)$ for a space $S$. A map $$\mathcal P : S\to\mathrm{Bun}_G(X)$$ is a (flat) family of principal $G$-bundles on $X$ parametrized by $S$, which means essentially a principal $G$-bundle $\mathcal P$ over $X\times S$. This is a "modular" interpretation of $\mathrm{Bun}_G(X)$; indeed, for $S=\mathrm{point}$, you retrieve the set (actually, groupoid) of principal $G$-bundles over $X$.
What about the étale, fppf, and other Grothendieck topologies? Well, I'm not 100% confident, but I think that if you work in the analytic setting you could disregard these abstractions and just work with the analytic (classical) topology, which I think is finer than each of the others (when seen as a Grothendieck topology, and compared appropriately with the others, working over $\mathbb C$).