TITLE: Understanding the definition of stacks
QUESTION [16 upvotes]: First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it here. 
I am trying to understand the notion of sheaves and stacks. Intuitively, sheaves are bit easy to understand as a gluing of compatible families of sets assign to opens sets of a topological space. In other words, it is a contravariant functor $\mathcal{F}:\mathbf{Open}(X)^{\operatorname{op}}\to\mathbf{Set}$ such that 
$$\mathcal{F}(U)=\lim\left(\prod_{i\in I}\mathcal{F}(U_i) \rightrightarrows \prod_{j,k\in I^2}\mathcal{F}(U_j\cap U_k)\right)$$
where $X$ is a topological space, $U$ is an open set of $X$ and $\{U_i\}_{i\in I}$ is any open cover of $U.$ Further, it is easy to imagine a sheaf as a étalé space over $X.$
Then I started reading about stacks using this notes and nlab as my primary sources. I learned that a stack is a contravariant functor $\mathcal{F}:\mathcal{C}^{\operatorname{op}}\to\mathbf{Grpd}$ satisfying a descent property and, categories fibered in groupoids over $\mathcal{C}$ is an intuitive way to think about stacks, where $\mathcal{C}$ is a site (category equipped with a coverage). Now I have following questions:

How can I understand the descent property for stack? To be more specific, how can triple fiber products (intersections) appear in the equalizer fork diagram?
What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks? Is this the correct analogue of étalé space of a stack?

REPLY [9 votes]: 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?
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:

for each index $i\in \Lambda$, an object $a_i$ in the category $\mathcal{F}(U_i)$,
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 $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$
(which you might have guessed already but let me say that), at the level of objects $$a\mapsto ((a|_{U_\alpha}),(\phi_{ij}))$$

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.

Now, you can ask what does equivalence of categories has anything to do with "sheaf like" properties? For a functor $\mathcal{D}\rightarrow \mathcal{C}$ to be an equivalence of categories, along other things, for each object $d\in \mathcal{D}$ we need an object $c\in \mathcal{C}$ such that, there is an isomorphism $F(c)\rightarrow d$.
Let $((a_\alpha),\{\phi_{\alpha\beta}\})$ be an object of $\mathcal{F}(\{U_\alpha\rightarrow U\})$. For this, by equivalence of categories, gives an element $a\in \mathcal{F}(U)$ that maps to $((a_\alpha),\{\phi_{\alpha\beta}\})$. That is, given an object $U$ of $\mathcal{C}$, an open cover $\{U_\alpha\rightarrow U\}$, for each collection of objects $\{a_\alpha\in \mathcal{F}(U_\alpha)\}$ that are compatible in some sense, there exists an object $a\in \mathcal{F}(U)$, such that, under appropriate restriction of $a$, you get the objects $a_{\alpha}$. This should remind the notion of sheaf on a topological space. This is how a stack is seen as a generalization of sheaf. 
References :

Notes on Grothendieck topologies, fibered categories and descent theory.
Orbifolds as stacks?
How is a Stack the generalisation of a sheaf from a 2-category point of view?