>What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks?

A category fibered in groupoids over $\mathcal{C}$ is given by a functor $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ satisfying certain conditions (I am not writing the definition as I assume you already know what is a category fibered in groupoids); look at Definition 4.2 in the paper [Orbifolds as stacks?][1]

Put a Grothendieck topology on the category $\mathcal{C}$; considering it as a site. 

Given an object $U$ of the category $\mathcal{C}$, we consider its fiber; a category, denoted by $\mathcal{F}(U)$, defined as 
$$\text{Obj}(\mathcal{F}(U))=\{V\in \text{Obj}(\mathcal{F}):\pi_{\mathcal{F}}(V)=U\},$$
$$\text{Mor}_{\mathcal{F}(U)}(V_1,V_2)=\{(f:V_1\rightarrow V_2)\in \text{Mor}_{\mathcal{F}}(V_1,V_2):\pi_{\mathcal{F}}(f)=1_U\}.$$

Given a cover $\{U_\alpha\rightarrow U\}$ of the object $U$ (remember that we fixed a Grothendieck topology), we consider its descent category, denoted by $\mathcal{F}(\{U_\alpha\rightarrow U\})$.
An object of the category $\mathcal{F}(\{U_\alpha\rightarrow U\})$ is given by the following data:
 
 1. for each index $i\in \Lambda$, an object $a_i$ in the category $\mathcal{F}(U_i)$,
 2. for each pair of indices $i,j\in \Lambda$, an isomorphism $\phi_{ij}:pr_2^*(a_j)\rightarrow pr_1^*(a_i)$ in the category $\mathcal{F}(U_i\times_{U}U_j)$

satisfying appropriate cocycle condition. 

Now, given a categroy fibered in groupoids $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$, an object $U$ of $\mathcal{C}$ and a cover $\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$ of $U$ in $\mathcal{C}$, there is an obvious functor $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\}).$$

> A category fibered in groupoids
> $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ is said to be a
> stack if, for each object $U$ of $\mathcal{C}$ and for each cover
> $\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$, the functor 
> $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$ is an equivalence of
> categories.

  [1]: https://arxiv.org/pdf/0806.4160.pdf