Coefficients in cohomology (Sorry if this is too elementary for this site)
I’m having some trouble understanding sheaf cohomology. It’s supposed to provide a theory of cohomology “with local coefficient”, and allow easy comparison between different theories like singular, Cech, de Rham and Alexander Spanier. What I don’t understand is: what’s all the fuss with coefficients that vary with each open set? Indeed what’s all the fuss with changing coefficients in an ordinary cohomology theory as in Eilenberg Steenrod?
Homology is trying to measure the “holes” of a space; wouldn’t integer coefficients suffice already? I’m not really sure what cohomology is trying to measure; at least I think the first singular group is trying to measure some kind of “potential difference”, like explained in Hatcher’s book. It gets worse for me when the coefficient group isn’t the integers. But when I get to sheaf cohomology I’m totally dumbstruck as to what it’s trying to measure, and what useful information of the space can be extracted from it. Now if it’s just about comparisons of different theories I can live with that…
Can someone please give me an intuitive explanation of the fuss with all these different coefficients? Please start off with why we even use different coefficients in Eilenberg Steenrod. Sorry if this is too elementary.
 A: A practical motivation: ordinary homology with, say, mod2 or rational coefficients are often easier to compute (and hence - to apply) than integral homology. 
A: This (elementary and perfectly standard) example might help show the power of sheaves with non-constant coefficients:
First, think about the circle $S^1$.  Suppose you want to understand (real) line bundles on the circle.  You can certainly cover the circle with two open contractible subsets $U_1$ and $U_2$ (which you can take to be the complements of the north and south poles), and we know that any line bundle on a contractible space is trivial.  So if you've got a line bundle $L$ over $S^1$, you can restrict it to either $U_i$ and get a trivial bundle $L_i$.  $L$ is built from these $L_i$ and the way they they are patched together over $U_1\cap U_2$.
Now what does it mean to patch the $L_i$ together over $U_{12}=U_1\cap U_2$?  It means choosing an isomorphism $L_1|U_{12}\rightarrow L_2|U_{12}$.  For any $x\in U_{12}$, the restriction of this isomorphism to the fiber $L_x$ over $x$ is an isomorphism between 1-dimensional vector spaces, and so (after choosing bases) can be identified with an element of ${\bf R}^*$ (the non-zero reals).  Therefore your patching consists of a continuous map 
$$U_{12}\rightarrow {\mathbb R}^*$$ 
which is to say, a Cech 1-cocycle for the sheaf of continuous ${\bf R}^{*}$-valued functions.
Now of course you could build a line bundle in some other way, say by starting with two different contractible sets $U_1$ and $U_2$.  When do two sets of patching data give isomorphic line bundles?  A little thought reveals that the answer is:  When and only when the corresponding cocycles give the same class in 
$$H^1(S^1,G^{*})$$ 
with $ G^{*} $ being the sheaf of continuous ${\bf R}^*$-valued functions.
Therefore line bundles are classified by $H^1(S^1,G^{*})$.  Now consider the exact sequence of sheaves
$$0 \rightarrow G \rightarrow G^*\rightarrow {\bf Z}/2{\bf Z}\rightarrow 0$$
where $G$ is the sheaf of continuous ${\bf R}$ valued functions, and the map on the left is exponentiation.  Follow the long exact sequence of cohomology, use the fact that $G$ is acyclic, and conclude that $H^1(S^1,G^*)=H^1(S^1,{\bf Z}/2{\bf Z})={\bf Z}/2{\bf Z}$.  In other words, there are exactly two real line bundles over $S^1$  --- and indeed there are:  the cylinder and the Mobius strip.
Exercise:  Do a similar calculation for ${\bf CP}^1$ (the Riemann sphere).  Conclude that the set of (complex) line bundles is in one-one correspondence with $H^2({\bf CP}^1,{\bf Z})={\bf Z}$.
A: As soon as you proceed from the first ideas of ''counting holes'' in a space to more advanced problems in algebraic topology, you will begin to appreciate local coefficient systems.
Even the passage from $Z$ to rings like $Z/2$ does not merely simplify computations, but allows you to detect more phenomena. For example, the map $RP^2 \to S^2$ that collapses $RP^1$ to a point is null in integral homology, but not in $Z/2$-homology. Think a few minutes about why this is not a contradiction to the universal coefficient theorem.
But local coefficient systems are useful in a variety of situations. Poincare duality for nonoriented manifolds has been mentioned (and in fact, it sheds light on the oriented case as well).
Then there is obstruction theory: If $f:X \to Y$ is a fibration with fibre $F$. Let $g:Z \to Y$ be a map. A basic problem of homotopy theory is to decide whether there can be a lift $h: Y \to X$ of $g$ through $f$. There is a sequence of obstructions to the existence of such a thing; and these obstructions live in $H^n (Z; \pi_{n-1}(F))$, but with twisted coefficients if $Y$ is not simply-connected.
Then the Leray-Serre spectral sequence comes to my mind: it relates the (co)homology of the base, the fibre and the total space of a fibration; and if the base isn't simply-connected, then local coefficients are inevitable.
Especially in the last two situations, the introduction of local coefficient systems makes the proofs more transparent even in the simply-connected case.
I admit that for most purposes of algebraic topology, the introduction of sheaves (more general than local coefficient systems) is overkill. The classical areas where sheaves are most important are complex analysis and algebraic geometry.
A: As Johannes Ebert says, the classical areas where sheaves are most
important are complex analysis and algebraic geometry.
There are two completely different kinds of sheaves one might consider
on a complex manifold: constructible sheaves (basically, locally
constant along a stratification) and quasicoherent sheaves (modules over
the ring of functions). It's kind of an amazing accident, and I think
rather misleading, that "sheaf theory" is useful for studying both
kinds of sheaves. Certainly there are theorems that apply to both kinds,
but most interesting theorems require you to assume one or the other. 
This is very much a matter of opinion, and I expect to get comments
disagreeing with me! Let me give just one example of what I mean.
For any sheaf at all, we can consider Cech cohomology using an open cover.
In the constructible-sheaf world (like you were getting in topology),
one likes to assume that the intersections of sets in the cover
are contractible, so all cohomology comes from gluing. 
In the quasicoherent-sheaf world, 
one likes to assume that the sets in the cover are affine
(and that the scheme is separated, so the intersections are likewise affine), 
again so all cohomology comes from gluing. Obviously one could
state a general theorem about acyclic covers or somesuch, but it's
crucial to bear in mind how different those are for the two
kinds of sheaves.
(N.B. Of course there are sheaves that are neither constructible
nor quasicoherent, and one occasionally does use, but not as often
as these two.)
A: I had the impression that Hatcher's book claims as motivation that local coefficients allow Poincaré Duality to work properly for non-orientable spaces.
