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:
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.