From my readings it is clear that schemes are the basis of algebraic geometry, and that sheaf cohomology is the main technical tool in algebraic geometry.
I am aware that the sheaf is fundamental to the definition of the scheme.
But I would really appreciate a clear explanation of why sheaf cohomology is so useful in studying schemes.
First: I'm by no means an expert on sheaf cohomology.
Lots of interesting geometry of a scheme $X$ can be understood by studying 'vector bundles' over $X$; i.e. one tries to analyze the category of (quasi-)coherent sheaves $\mathcal{QCoh}(X)/\mathcal{Coh}(X)$ on $X$.
Personally I think of this as an analogon of how one often studies $R$-modules, when one is interested in the ring $R$. For example, if $X = \mathrm{Spec} A$ is affine, one has an equivalence of categories of $\mathcal{Mod}_A \cong \mathcal{QCoh}(X)$. So.. like one glues ("geometrized" versions of) rings to obtain schemes, one glues ("geometrized" versions of) modules over those rings to obtain quasicoherent modules over those schemes. The coherent case furthermore restricts some finiteness conditions (geometrically I think this means restricting to "finite-rank" vector bundles). (Vector bundles correspond to locally free sheaves (of finite rank); one can also think of $\mathcal{QCoh}(X)/\mathcal{Coh}(X)$ as the smallest abelian subcategories of the category of sheaves of $\mathcal{O}_X$-modules that contains (things equivalent to) vector bundles/finite rank vector bundles; I'm not sure if it's possible to "formalize" this statement though.)
Now, sheaf cohomology seems to be a very valuable tool to study (quasi-)coherent modules over a scheme X. As you probably know, the global section functor $\Gamma(X,-)$ is left-exact, but not in general right-exact. As sheaf cohomology is the (right-)derived functor of $\Gamma(X,-)$ it measures how far global sections are from being surjective.
I think this is quite a natural question:
If $ \varphi:\mathcal{F} \rightarrow \mathcal{G}$ is surjective this just means that locally, the given morphism is surjective, i.e. given any section $s \in \Gamma(X,\mathcal{G})$ and any point $x\in X$ one know that there exists an open subset $x \in U^x \subseteq X$ s.t. $s|_{U^x} = \varphi_{U^x}(t^x)$ for some $t^x \in \Gamma(U^x,\mathcal{F})$.
Now one might start to think about the following: When is it possible to choose the $\{t^x\}_{x\in X}$ s.t. they glue to a global section $t \in \Gamma(X,\mathcal{F})$ (which then satisfies $\varphi_X(t)=s$, as one easily verifies).
Tinkering a little bit with that question you might realize, that the existence of such a global section $t$ is equivalent to the vanishing of a certain cohomology class in $H^1(X, \mathrm{ker}\varphi)$ defined by the given data - namely: $0 \rightarrow \Gamma(X,\mathrm{ker}\varphi) \rightarrow \Gamma(X,\mathcal{F}) \rightarrow \Gamma(X,\mathcal{G}) \rightarrow H^1(X, \mathrm{ker}\varphi) \rightarrow \cdots$ is exact and $s \in \Gamma(X,\mathcal{G})$ is in the image of $\varphi_X$ iff. it is in the kernel of $\Gamma(X,\mathcal{G}) \rightarrow H^1(X, \mathrm{ker}\varphi)$.
So sheaf cohomology serves a very "down-to-earth" purpose, namely measuring the obstruction of the existence of sections satisfying certain local conditions. This concrete purpose however is combined with the "abstract machinery" of homological algebra to obtain "better behaved" algebra - long exact sequences etc. (It is not easy/clear how to interpret higher cohomology groups though as far as I know)
Also note that lots of interesting geometric questions are at least related to being able to conclude things globally, if they are known locally. So I think it's hardly surprising that sheaf cohomology is a very valuable tool in algebraic geometry (especially as other techniques commonly used in differential geometry - like partitions of unity - aren't really there in algebraic geometry as the algebro-geometric objects are too "rigid" in some sense).
Maybe others can give you more concrete examples, this is only my current understanding of the situation (which is far from perfect).