Consider the category of manifolds $\text{Man}$.

A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the category $\text{Man}$) for this $\mathcal{G}$, the stack of principal $\mathcal{G}$ bundles, denoted by $B\mathcal{G}$. Any stack which is isomorphic to $B\mathcal{G}$ is called a geometric stack. So, geometric stacks are same as Lie groupoids; the **”geometric” stacks of interest over the category of manifolds** are precisely the **stacks associated to groupoid objects in the category of manifolds**.

Now, fix a scheme $S$ and consider the category, $\text{Sch}/S$, of schemes over $S$. Fix some Grothendieck topology (I do not know many, so, you can fix what ever is comfortable). In this category also I can talk about groupoid objects, take one such and denote by $\mathcal{G}$. Is there a way to associate a stack over the category $\text{Sch}/S$.

Is it the case, similar to the case of manifolds, that the **”geometric” stacks of interest over the category $\text{Sch}/S$** are precisely the **stacks associated to groupoid objects in the category of $\text{Sch}/S$**? I mean to ask are Algebraic stack precisely the stacks coming from groupoid object in category $\text{Sch}/S$? I guess the answer is No (please feel free to add some comments regarding this).

Are stacks over $\text{Sch}/S$ associated to groupoids over $\text{Sch}/S$ of any interest? Do they cover “most” of Algebraic stacks over the stack $S$ with what ever topology on $\text{Sch}/S$? Are there (basic) examples of stacks associated to groupoids over $\text{Sch}/S$ that are not algebraic?

knewit was a question by you! :-) If you allow your "geometric" groupoids to be internal toalgebraic spacesinstead of just schemes, and consider thelisse(smooth) topology, then I think all the answers become "yes" by Laumon--Moret-BaillyChamps algébriques, (4.3) and Proposition (4.3.1). $\endgroup$10more comments