Let $C$ be a category. If $X$ is an object of $C$, then write $\underline{X} = \text{Hom}(-, X): C^{\text{op}}\to \text{Sets}$ for the corresponding functor.
Suppose that a "cover" of $X$ is a subfunctor $i: \mathcal{U}\to \underline{X}$ such that, for every object $Y$ of $C$, the function \begin{equation} i^\ast: \text{Hom}(\underline{X}, \underline{Y})\to \text{Hom}(\mathcal{U}, \underline{Y}). \end{equation} is a bijection. To me this definition seems to perfectly capture the notion of a cover: each morphism $\mathcal{U}\to \underline{Y}$ uniquely defines a morphism $X\to Y$. What more (or less) could you ask of a cover?
I have never seen this definition stated or discussed anywhere. I am guessing this notion of cover is at least somewhat related to a covering sieve for the canonical topology (but I'm not sure how to prove it), because it seems that the above definition is the minimal requirement for every representable functor to be a sheaf.
Now the category of schemes, for example, has many different types of covering sieves, leading to many different Grothendieck topologies. Why is my above notion of cover not sufficient for all purposes? What is a specific example of something you can do with insert-grothendieck-topology-here that you wouldn't be able to do with my above definition of cover?