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.