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}
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.
Example  Single Element SetA very simple example would be where C^{op} is a single element set (terminal object in set). Hom( C^{op}, Set) therefore contains set of single arrows, one for every element of the set. 

Example  GraphHere C^{op} is a category with two objects E (for edge) and V (for vertex) also two arrows s (for source) and t (for target). This allows us to build a structure on top of set where the diagram on the right commutes. We can therefore build up complex graphs from individual vertices and edges. 

Example  Relational DatabaseHere C^{op} is a database schema. This imposes a structure on the sets which are the database tables. This implements a category of simplical databases. 