A concrete site is a site whose objects can be thought of as sets with extra structure: it is a category that is a concrete category and a site in a compatible way.
In a category of presheaves on a concrete site one can consider concrete presheaves.
A concrete site is a site $C$ with a terminal object $*$ such that
the functor $Hom_C(*,-) : C \to Set$ is a faithful functor;
for every covering family $\{f_i : U_i \to U\}$ in $C$ the morphism
is surjective.
every (small) concrete category becomes a concrete site when equipped with the trivial coverage (every covering family consists of just an identity morphism);
Any small subcategory of concrete categories such as Top, Diff, etc, with their standard coverages (by open cover)s;
for instance CartSp (covering families are open covers);
which is the site for smooth spaces, including diffeological spaces.