Applications of the "other" definition of sheaves In most literature, when you try to look for the definition of sheaves you will see the usual definition for presheaves as a functor from a topological space (or from a Grothendieck topology) to some category and then sheaves would require this category to be complete and you have some exactness/equalizer condition. 
But then for some categories there is another equivalent definition. You are defined a "protosheaf" (there are various names for these creatures), a sheaf space, a base space, a local homeomorphism between the sheaf space and the base space, you are even already defined a stalk.. but this definition seems not to be very abstract in the category-theoretical point of view as I only see this kind of definition for very specific categories (for instance in the category of groups or rings, you want the addition operation defined on the fiber product of the sheaf space over the base space to be continuous). What is the equivalent category theoretical way of defining a sheaf using this method? In which cases does this definition give us a more psychological advantage than the aforementioned one? I have personally found the former definition more advantageous in my practice, but there are some mathematical practices by which the latter definition might be more useful.
 A: There is a generalization of a presheaf called a "fibered category" or a "grothendieck fibration".  This is analogous to the etale space construction for presheaves on O(X).  Every presheaf in the sense of a presheaf taking values in Sets (most other constructions come from enriching presheaves of sets) can be identified with a very simple type of fibered category.  In general, fibered categories wth a fixed cleavage (something like a skeleton of pullbacks) define a contravariant pseudofunctor taking values in the 2-category of categories.  It is only a pseudofunctor because composition is not in general strictly associative, merely associative up to unique isomorphism.  You should check out Vistoli's book on descent, fibered categories, and grothendieck topologies here: http://homepage.sns.it/vistoli/descent.pdf .  
But to answer your question, to generalize the etale space for sheaves, you'll have to introduce the idea of descent for 1-stacks, then sheaves become degenerate stacks, i.e. 0-stacks.  If you're trying to deal with sheaves without resorting to too much category theory, you have to remember that presheaves of abelian groups, for example, are presheaves of abelian group objects in Sets.  All of the categories that you've mentioned are monadic with respect to the forgetful functor adjunction with Sets, so we can always just take objects of that sort in the category of sets.  It's the reason why a "presheaf of topological spaces" doesn't really make a lot of sense, since Top is not algebraic over Sets.  So there is no good way to define sheaves taking values in an arbitrary category without substantially increasing the generality.
If you aren't familiar with what I'm talking about specifically, Mac Lane's "sheaves in geometry and logic" has a very detailed explanation of how the etale space works, and how we can produce sheaves that take values in algebraic categories, but not arbitrary categories.  It also has a very in-depth construction of the etale space for presheaves of sets and also proves the equivalence of categories between the full subcategory of sheaves of sets and the full subcategory of "bundles" (Mac Lane's terminology here, so don't get confused when he calls the etale space the etale bundle) called etale spaces.
A: There is an important application of the "other definition" in arithmetic geometry: It is used to give a canonical action of the frobenius on sheaf defined over a finite field:
There is an equivalence of categories between constructible sheaves ("usual sheaves") on a variety $X$ and algebraic spaces etale over $X$ ("other sheaves"). 
Now one can use the frobenius action on spaces and translate it back through the equivalence into an action on sheaves.
A: My understanding is that what you're talking about is the espace etale (not sure where accents go offhand) of the sheaf, and was the original definition.  Proving that the sheaf of sections of this space is an exercise in Hartshorne.  The only place I've ever seen that definition used seriously is in this complex analysis book by Narasimhan and Nievergelt.  Though they use the espace etale definition exclusively, and for pretty much the whole book to handle germs of holomorphic functions.
A: Here is an elegant application of sheaves seen as étalé spaces.
Consider a complex manifold $M$. It automatically comes with a holomorphic local isomorphism $\pi: \mathcal O_M \to M $ described as follows. As a set $\mathcal O_M$ is the set of all germs of holomorphic functions at all points of $M$.The map $\pi$ sends a germ to the point at which it is considered. Then we endow $\mathcal O_M$  with the following topology. For an open connected set $U\subset M$ and a holomorphic function $f$ on $U$, denote by $[U,f]\subset\mathcal O_M$ the set of all germs $f_a$ with $a\in U$.
These $[U,f]$ are decreed to be an open basis for the topology of $\mathcal O_M$.Then there exists a unique complex structure on $\mathcal O_M$ such that $\pi: \mathcal O_M \to M $ becomes a HOLOMORPHIC local isomorphism. On $\mathcal O_M$ there lives a universal tautological holomorphic function
$F:\mathcal O_M \to \mathbb C: f_a \to f_a (a)$. (Note that $\mathcal O_M$ is huge, disconnected but Hausdorff).
And now for the punchline : given a holomorphic function $f$ on $U\subset M$, take the connected component $Riem(U)$ of $[U,f]$ in $\mathcal O_M$. Together with the restriction $F|Riem(U)$,  this is the maximal holomorphic extension of $f$: a sophisticated concept admirably handled by sheaves as étalé spaces  (The manifold $Riem(U)$ is called the domain of existence of $f$.)
Even in dimension one and for $M=\mathbb C$ this is quite powerful: you get the Riemann surface $(Riem(U), F|Riem(U))$ of any holomorphic function $f$ on an arbitrary domain $U\subset \mathbb C$
without the cutting, pasting, continuation along paths,... of which classical books on complex analysis are so fond.
A reference for this might be Fritzsche-Grauert's book "From Holomorphic Functions to Comples Manifolds", Chapter II, $$8,9 (Springer, GTM 213). The book by Narasimhan and Nievergelt that Charles so pertinently and quickly evoked  seems to handle the dimension one case (which actually suffices to convey the sheaf idea).
Finally, it is noteworthy that the EGA-style definition that Hartshorne gives for the structure sheaf $\mathcal O$of the affine scheme $Spec(A)$ (page 70 of THE BOOK) is exactly analogous to the description above: the étalé space is the disjoint union of the all the local rings $A_P$ for $P\in A$ and $\mathcal O(U)$ is the set of continuous maps of $U$ into the étalé space;
Only, Hartshorne doesn't say what the topology is on the étalé space and the continuity condition is replaced by an ad hoc description in terms of elements of the rings of fractions $A_f$.
A: The "sheaf space = espace étalé" definition is better (opinions may vary) than the definition by specifying sections over open sets in at least the following cases:


*

*Working with constant or near constant sheaves, such as constructible sheaves.

*To define the restriction $F|_S$ to an arbitrary, not necessarily open, subset $S\subset X$, and in particular to understand the set of sections $F(S)$ over a non-open subset.

*To define the pullback $f^{-1}F$.

*To prove that $f_*$ and $f^{-1}$ are adjoint.


Since the two definitions are equivalent, all of these can be accomplished by using only open sets, but using the sheaf space gives a better geometric picture of the situation. 
These examples apply to sheaves over topological spaces, not necessarily to the other categories you mentioned.
