The notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\H^0(X,\HH^p(X,F))$ appears to be a notation for the $p$th (hyper)cohomology group of $F$, i.e., take the injective resolution (equivalently: local fibrant replacement) of $F$ and then take the $p$th cohomology group of its sections over $X$. Thus, $\HH^p(X,F)$ could be taken to be the presheaf that sends an open subset $U⊂X$ to the $p$th hypercohomology group of $F$ over $U$. The global sections of this presheaf are precisely the $p$th hypercohomology group of $F$ over $X$.

The keyword to look for in older sources is hypercohomology. The less old sources may simply be talking about the derived category of sheaves. The newer sources might say something like the “homology presheaf of an ∞-sheaf of chain complexes”.

As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in Étale Cohomology as the $p$th right derived functor of the inclusion functor from the category of sheaves of abelian groups to the category of presheaves of abelian groups on the same site. This is also known as hypercohomology and is studied under this name by Grothendieck, although I could not locate this notation in his paper.

In modern terms, we can compute $\HH^p(X,F)$ by replacing $F$ with a locally quasi-isomorphic presheaf $G$ of chain complexes that satisfies homotopy descent, e.g., by fibrantly replacing in the local injective model structure on presheaves or sheaves of chain complexes, or the local projective model structure on presheaves of chain complexes. Then $\HH^p(X,F)$ is simply the $p$th cohomology group of $G$ computed in the abelian category of presheaves of abelian groups.

The keyword to look for in older sources is hypercohomology, the newer sources may simply be talking about the derived category of sheaves, and even newer sources might say something like the “cohomology presheaf of an ∞-sheaf of chain complexes”.

$\def\HH{\underline{\rm H}}\HH^p(X,F)$ is just the $p$th homology sheaf of the resolution of $F$, i.e., we do not take global sections here. Thus: take the injective resolution (equivalently: local fibrant replacement) of $F$ and then take its $p$th homology sheaf, which is a sheaf of abelian groups on $X$.

The keyword to look for in older sources is hypercohomology. The less old sources may simply be talking about the derived category of sheaves. The newer sources might say something like the “homology sheaf of an ∞-sheaf of chain complexes”.

The notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\H^0(X,\HH^p(X,F))$ appears to be a notation for the $p$th (hyper)cohomology group of $F$, i.e., take the injective resolution (equivalently: local fibrant replacement) of $F$ and then take the $p$th cohomology group of its sections over $X$. Thus, $\HH^p(X,F)$ could be taken to be the presheaf that sends an open subset $U⊂X$ to the $p$th hypercohomology group of $F$ over $U$. The global sections of this presheaf are precisely the $p$th hypercohomology group of $F$ over $X$.

The keyword to look for in older sources is hypercohomology. The less old sources may simply be talking about the derived category of sheaves. The newer sources might say something like the “homology presheaf of an ∞-sheaf of chain complexes”.