under construction
Generally, for $X$ an object we think of as a space, a cover of $X$ is some other object $Y$ together with a morphism $\pi : Y \to X$, usually an epimorphism demanded to be well behaved in certain way.
The idea is that $Y$ provides a “locally resolved” picture of $X$ in that $X$ and $Y$ are “locally the same” but that $Y$ is “more flexible” than $X$.
The archetypical example are ordinary covers of topological spaces $X$ by open subsets $\{U_i\}$: here $Y$ is their disjoint union $Y := \coprod_i U_i$.
More generally, you might need a cover to be family of maps $(\pi_i: Y_i \to X)_i$; if the category has coproducts that get along well with the covers, then you can replace these families with single maps as above—see superextensive site.
In the context of sheaf and topos theory a cover on an object $U$ in a category $C$ is a collection of morphisms $\{U_i \to U\}_{i \in I}$.
A specification of a collection of covers for each object of the category, subject to some compatibility condition, makes a coverage on $C$. If the collection of covers in a coverage is being closed under some operations, the result is called a Grothendieck topology. Equipped with a coverage/Grothendieck topology, the category is called a site. See there for more details.
Covering families $\{U_i \to U\}$ in $C$ have incarnations as single morphisms in the category of presheaves $PSh(C)$ over $C$, and these are also sometimes called covers :
the Cech nerve of the morphism $\coprod_i U_i \to U$ in $PSh(C)$ is a simplicial object in $PSh(C)$
Its colimit is the local epimorphism on $U$ that is the incarnation of the covering family in $C$, now in $PSh(C)$.
In higher category theory, when we do not restrict to presheaves, for instance when we use simplicial presheaves, the full Cech nerve itself $C(\{U_i\}) \to U$ is the “local epimorphism”, the covering map.
More generally, given a coverage one can form hypercovers in the category of simplicial presheaves, by starting with a Cech nerve and then iteratively refining it in each degree further and further by more covers.
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In the category $C$ =Top of topological spaces or $C$ = Diff of smooth manifolds or similar, one has the notions
open cover – for $X \in C$ a space, an open cover is a collection $\{U_i \subset X\}$ of open subsets, that cover $X$ in the obvious naive sense of the word, i.e. which are such that their union equals $X$;
good cover – a cover $\{U_i \to X\}$ is called a good cover (or good open cover if in addition it is an open cover) if all of the $U_i$ and all their finite intersections $U_{i_1} \times_X U_{i_2} \times_X \cdots \times_X U_{i_n}$ are contractible as topological spaces.
A parameterized version of this is a stacked cover.
There is also the notion of
of a topological space or manifold. This is a priori an independent notion of cover, but for the standard Grothendieck topologies on Top, Diff, etc. the projection $\{\hat X \to X\}$ from a covering space is a covering family.