First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a quasi-coherent sheaf on each $U_i$ and $\phi_{ij}$ is an isomorphism $pr_2^*\mathcal{E}_j\to pr_1^*\mathcal{E}_i$ in $Qcoh(U_{ij})$ which satisfies the cocycle condition $$ pr_{13}^*\phi_{ik}=pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}. $$ We can make the descent data into a category where the morphisms are those compatible with the $\phi_{ij}$'s. Moreover we have an equivalence of categories $$ \text{Desc}(X,\mathcal{U})\simeq Qcoh(X). $$

Of course the definition of descent data has various generalizations. For example instead of the category of quasi-coherent sheaves we can consider an arbitrary fibered category $\mathcal{F}$ over spaces. For any map between spaces $\pi: U\to X$ we have a cosimplicial diagram of categories $$ \mathcal{F}(X)\to \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$ An object with descent data on is a collection $(\mathcal{E},\phi)$ where $\mathcal{E}$ is an object in $\mathcal{F}(U)$ and $\phi$ is an isomorphism $pr_2^*\mathcal{E}\to pr_1^*\mathcal{E}$ which satisfies the cocycle condition $$ pr_{13}^*\phi=pr_{12}^*\phi\circ pr_{23}^*\phi. $$ We notice that the category $\text{Desc}(U\to X)$ is equivalent to the fiber product of categories $\mathcal{F}(U)\times_{\mathcal{F}(U\times_X U)} \mathcal{F}(U)$ (as Zhen Lin points out in the comment, we need to also consider $\mathcal{F}(U\times_X U\times_X U)$).

We also notice that if $\pi$ is flat then $\phi$ can be also expressed as the comodule structure $\mathcal{E}\to \pi^*\pi_*\mathcal{E}$ since $\pi^*\pi_*\mathcal{E}\cong (pr_2)_*pr_1^*\mathcal{E}$. Moreover in the flat case (together with some condition on $F$ I guess) we have $$ \text{Desc}(U\to X)\simeq \mathcal{F}(X). $$

Now we consider the case that $\mathcal{F}$ contains some higher structure. In this case we have an augmented cosimplicial diagram of (higher) categories $$ \mathcal{F}(X)\to \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$ and the definition of descent data should be modified. For example in the recent version of Yekutieli's Twisted Deformation Quantization of Algebraic Varieties Section 5, an explicit definition of descent data of cosimplicial cross groupoid is given.

Before given the explicit construction, I would like to know what SHOULD the descent data be. Or more precisely what are the properties that descent data must satisfy?

One attempt is to say that descent data are extra structures (say comodule) on $\mathcal{F}(U)$ such that we have the (weak) equivalence $\text{Desc}(U\to X)\simeq \mathcal{F}(X)$, but the equivalence only exists in good (say flat) cases while the descent data can be given in general cases.

Therefore is the alternative description better? The category of descent data should be the homotopy limit (or totalization) of the cosimplicial diagram $$ \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$