fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation.
[2] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.
The small fppf site of X is the category O(Xfppf) whose objects are schemes U with a fixed morphism U → X which is part of some covering family.
"Fppf" is an abbreviation for "fidèlement plate de présentation finie", that is, "faithfully flat and of finite presentation".
The definition of the fppf pretopology can also be given with an extra quasi-finiteness condition; it follows from Corollary 17.16.2 in EGA IV4 that this gives the same topology.
The small fpqc site of X is the category O(Xfpqc) whose objects are schemes U with a fixed morphism U → X which is part of some covering family.
The procedure for defining the cohomology groups is the standard one: cohomology is defined as the sequence of derived functors of the functor taking the sections of a sheaf of abelian groups.
While such groups have a number of applications, they are not in general easy to compute, except in cases where they reduce to other theories, such as the étale cohomology.
The following example shows why the "faithfully flat topology" without any finiteness conditions does not behave well.