Perhaps because I find the algebraic definition of étale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of étale morphisms of topological spaces. In topological spaces, we have an equivalence of categories $\acute Et\downarrow X\cong\text{Sh}(X)$, the category of étale spaces over $X$ is equivalent to sheaves on $X$. This probably determines the étale topology uniquely, and seems to me a good way to motivate the definition. Then I would work out its description in terms of open sets.
Can I approach the definition of étale morphisms of schemes the same way? What is the category of étale schemes over scheme $X$ equivalent to? I suppose it is not sheaves on (the underlying topological space of) $X$.