A 'presheaf' category is a special case of a functor category (see page here). It is a contravarient functor from a category 'C' to Set.
Since it is contravarient it is usually written:
C^{op}>Set
or
Set^{Cop}
So we have a functor from a category to the category of sets. This sends objects to sets and arrows to functions in a way that plays well with the objects. In category theory we don't tend to look inside objects but in set theory we look inside sets. S how can we define these morphisms? In order to work with sets it needs to preserve something to do with subsets.
Presheaf Category
In a presheaf category the object is a functor. 

Morphisms are structure preserving maps between these functors. 
In the theory of topological space a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
Presheaf Examples
 A simplicial set is a presheaf on the simplex category
 A globular set is a presheaf on the globe category.
 A cubical set is a presheaf on the cube category.
Examples  Graph and Simplical ComplexesSee page here. 

Example  Relational DatabaseSee page here. 
Further Information
For a more general introduction to sheaves see the page here.