Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$.
Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i f_*{\mathcal{F}}$ which is also a sheaf on $Y$.
Moreover, we have another functor $\underline{\mathbf{R}} f_*$ which is the derived functor between derived categories of complex of sheaves (here $\mathcal{F}$ viewed as a complex in degree $0$):
$$\underline{\mathbf{R}} f_* : D(X) \to D(Y).$$
Then what is the usual meaning of $\underline{\mathbf{R}}^i f_*(\mathcal F)$? Here are two possible understandings:
(1) $\underline{\mathbf{R}}^i f_*(\mathcal F)$ is a SHEAF which is the $i$-th cohomology of $\mathbf{R} f_*(\mathcal F)$ (I feel that this is the common convention)
(2) Using the functor $\mathbf{R}^i f_*$ (this is not the left exact functor) to define the derived functor between derived categories $D(X) \to D(Y)$. Hence $\underline{\mathbf{R}}^i f_*(\mathcal F)$ is a sheaf of complex.