One last post this year. I came across the following statement in the Stacks project:

[...] describing the fppf topology as being equal to the topology "generated by" Zariski coverings and by coverings of the form $\{f\colon T\to S\}$ where $f$ is surjective finite locally free.

This fact is already discussed on the Stacks project blog. Recall that it should be interpreted as "the categories of sheaves for this topology are the same". A similar phenomenon happens for étale and smooth.