In a functor category the object is a functor. 

Morphisms are structure preserving maps between these functors. 
Since the objects are all functors from X to Y we can redraw it like this: 
So we can the that these morphisms are really natural transformations (as described on page here).
So a functor category is a category where:
 The objects are functors.
 The morphisms are natural transformations.
Example of a Functor Category
See presheaves on page here.