Relative version of sheaf cohomology?

To elaborate on Kevin's comment: If $f: X to S$, and $mathcal F$ is a sheaf on
$X$, then $f_*mathcal F$ is the sheaf on $S$ defined by $H^0(U,f_*mathcal F)
:= H^0(f^{-1}(U),mathcal F).$

Taking the derived functors of $f_*$ gives functors $R^if_*$, and it turns out
(fairly easily) that $R^if_*(mathcal F)(U)$ is the sheaf associated to the presheaf
$U mapsto H^i(f^{-1}(U),mathcal F)$. If $i > 0,$ then this presheaf may not be
a sheaf (unlike the $i = 0$ case), and this is related to the fact that it can be a little subtle to compute the stalks of $R^if_*mathcal F$ in general; for example, it need not
be the case in general that the stalk $(R^if_*mathcal F)_s$ is equal to
$H^i(f^{-1}(s),mathcal F)$. (E.g. think about the case when $f$ is the inclusion of
a punctured disk into a disk, $mathcal F$ is the constant sheaf ${mathbb Z}$,
and $s$ is the centre of the disk (so that $f^{-1}(s)$ is empty).)

In other words, $R^if_*mathcal F$ does not always literally interpolate the cohomology
of the fibres.

There is one case where one knows that $R^if_*mathcal F$ does interpolate
the cohomology of the fibres: if the map $f$ is proper, than the proper base-change theorem
says that the stalk of $R^if_*mathcal F$ at $s$ is the cohomology of $mathcal F$ along
the fibre of $s$. (One good place for these kinds of facts is the beginning of Borel's book on Intersection Cohomology.)

Also, in the context of maps of varieties, if $f$ is proper and $mathcal F$ is a coherent
sheaf, then the completed stalk of $R^if_* mathcal F$ at $s$ coicides with the cohomology
of the pull-back of $mathcal F$ to the formal completion of $f^{-1}(s)$ in $X$. (This is Grothendieck's proper base-change theorem, proved in some form in Hartshorne, Ch. III.)