Endofunctors

What is the difference between an endofunctor and a functor?

An endofunctior has the same domain and codomain while a functor might have different codomain from domain. An endofunctor is a functor.

An endofunctor is a functor that maps a category to itself.

In Haskell, all Functors are endofunctors, because they map Haskell types and functions to new Haskell types and functions. For example, Maybe : Hask → Hask maps each type a to Maybe a, and each function a -> b to a function Maybe a -> Maybe b. (How this is done is exactly what the definition of fmap tells you.)