The term "forgetful functor" is not perfectly well defined. Depending on context, I've seen it defined as "faithful functor with a left adjoint", because most notions of "Forget" should have a corresponding notion of "Free".
Edit: I should emphasize that there are many notions of "forgetful functor", and it is not a canonically-defined word. J. Baez has thought about what the right notions of "forget" are. For him, a functor "forgets at most properties" if it is full and faithful, "forgets at most structure" if it is faithful, and "forgets at most stuff" if it is (empty list). I think it is not necessarily true that a full and faithful functor has a left adjoint.