from what little i have looked into category theory, the yoneda functor is the embedding of any category into the category of functors and natural transformations. The are 2 variants of the yoneda functor, covariant and contravariant which takes the category to a contravariant and covariant hom functor respectively. This is the yoneda embedding, which is a special case of the yoneda lemma. Basically its telling you that the objects and morphisms of any category is naturally isomorphic to the functor category where objects are set valued functors (functors taking C to the category of Sets where C is any category) and morphisms are natural transformations.
for example group/monoid actions, one can think of them as group/monoid homorphisms between the group/monoid and the symmetry group. The Yoneda functor is the generalized version this concept, taking the category (the category of the group/monoid) to the category of functor