I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^\* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^\*) $?



Cohomology of complexes
There are two concepts defined for complexes of sheaves, both called "cohomology", which are related but different. The more *basic* concept is the kind of cohomology that is defined for any complex $A^\bullet$ of objects in an abelian category:
$$\DeclareMathOperator{\im}{im}H^n(A^\bullet) = \ker d^\bullet / \im d^{\bullet - 1},$$
where $d^\bullet$ are the differentials. This is not the kind you're asking about. What you are seeing is the *derived functor* kind of cohomology, in this case the derived functor of global sections of sheaves. If $X$ is a topological space (or site) and $\def\sh#1{\mathcal{#1}}\sh{F}^\bullet$ is a complex of sheaves on $X$, then:
$$H^n(X, \sh{F}^\bullet) = H^n \bigl(R\Gamma(X, \sh{F}^\bullet)\bigr).$$
The $H^n$ on the right is the "complex cohomology" and the $H^n$ on the left is what we are defining to be the "sheaf cohomology". Here, $R\Gamma(X, -)$ is the functor obtained in the manner described by Sandor in his answer: find (any) quasi-isomorphism of $\sh{F}^\bullet$ with a complex of *injective* sheaves $\sh{I}^\bullet$, and define:
$$R\Gamma(X,\sh{F}^\bullet) = \Gamma(X,\sh{I}^\bullet);$$
in other words, just apply global sections directly to the complex of injectives. For why this works, and what it means, you'll need to read a book; Weibel is a great reference, though I really like Gelfand–Manin for learning. YMMV.


