If f is a Functor then the free Monad on f is the type of trees whose nodes are labeled with the constructors of f. The word "free" is used in the sense of "unrestricted" rather than "zero-cost": Free f makes no constraining assumptions beyond those given by f and the definition of Monad. As used here it is a standard term from the mathematical theory of adjoint functors.
Cofree comonads are dual to free monads. They provide convenient ways to talk about branching streams and rose-trees, and can be used to annotate syntax trees. The cofree comonad can be seen as a stream parameterized by a Functor that controls its branching factor.