Briefly on the purpose of Functors, Applicatives and Monads

In a recent thread on /r/haskell about how to motivate the AMP proposal in teaching, I read a comment that finally helped me understand the purpose of Functors, Applicatives and Monads.

Wait, what is AMP?

For the reader that hasn’t followed this debacle, the AMP proposal is basically about making Applicative a superclass of Monad. That this hasn't been so is often considered one of the historical errors Haskell carries around, since a Monad is always an Applicative.