TITLE: What is an intuitive Geometrical explanation of a "sheaf?"
QUESTION [25 upvotes]: As I understand it, a sheaf is a very broad concept, but is most often used when referencing a function that maps algebraic structures (like rings) to points on a manifolds. 
Is a sheaf just yet another type of function, but built specifically for algebraic geometry? What is so special about it? 
I have seen this question before on other sites, but most of the answers are way too long & non-intuitive.

REPLY [4 votes]: A sheaf $F$ on a topological space is something that associates to every open set an object $F(U)$, e.g. an abelian group. Elements of $F(U)$ are usually called sections on U.
Furthermore,whenever you have $U \subset V$ there is an operator of restriction $| U: F(V) \to F(U)$. As the name suggests, you want that whenever $U \subset V \subset W$, if you first restrict to V and then to U you obtain the same result as if you directly restrict to U. The properties you require for operators of restriction depend on the structure on the objects you chose. For example, in case of abelian groups, they must be homorphisms. 
With these data you get a presheaf. A sheaf has the additional glueing property, which is the following. Take an open $U$, and an open covering $U_{a}$ of $U$. Denote by $U_{ab}=U_a \cap U_b$. Now assign sections $s_a$ over $U_{a}$ such that $s_a | U_{ab} = s_b | U_{ab} $, i.e. they coincide on intersections. Then exist a unique $s$ over U which extends the $s_a$, i.e. $s | U_a = s_a $.
The simplest example is the following: associate to every open $U$ of a manifold, e.g. the 2-dimensional sphere,  the set of $C^{\infty}$ functions from $U$ to $\mathbb{R}$. The restriction is the usual restriction, which furthermore preserve pointwise multiplication and addition. The glueing property is easily verified.
Let me now explain why you have heard of sheaves as "special functions". Indeed, suppose you want to define a "multifunction", for example $y= \pm \sqrt{1-x^2}$. It is not so comfortable to deal with them, so we make this trick.
Consider $\pi : X \times Y \to X$ which send $(x,y) \to x$. Convince yourself that assigning a function continous $f:X \to Y$ is the same as assigning a continuous section $s$ of $\pi$, i.e. $\pi s =1$.
In the example above, consider the subset $T \subset \mathbb{C} \times \mathbb{C}$ given by $T=\{x^2+y^2=1\}$ equipped with the projection $\pi$ on $x$. Now consider the sheaf $F(U) = \{s: U \to T: \pi s = 1\}$ of continous sections. You can verify the above axioms. Indeed, this sheaf has two different sections over every open set, which correspond to the choice of the sign. Thus it is in a certain sense a "multifunction".